WORKS  OF  PROF.  H.  W.  SPANGLER 

PUBLISHED  W 

JOHN  WILEY  &  SONS. 


Valve-Gears 

Designed  as  a  Text-book  giving  those  parts  of  the 
Theory  of  Valve-gears  necessary  to  a  clear  under- 
standing of  the  subject.  8vo,  xii  +  17^  pages,  109 
figures,  cloth,  $2.50. 

Notes  on  Thermodynamics. 

The  Derivation  of  the  Fundamental  Principles  of 
Thermodynamics  and  their  Application  to  Numer- 
ical Problems.  Fourth  Edition,  izmo,  vii+7S 
pages,  24  figures,  cloth,  $1.00. 

Elements  of  Steam  Engineering. 

By  H.  W.  SPANGLFR,  Whitney  Professor  of  Dynam- 
ical Engineering  in  the  University  of  Pennsylvania, 
ARTHUR  M.  GREENE,  Jr.,  Professor  of  Mechanical 
Engineering  in  the  University  of  Missouri,  and  S.  M. 
MARSHALL,  B.S.  in  E.E.  Second  Edition,  Revised 
and  Enlarged.  8vo,  v  +  297  pages,  290  figures, 
cloth,  $3.00. 


VALVE-GEARS 


BY 

H.  W.  SPANGLER, 

Whitney  Professor  of  Mechanical  Engineering  in  the  University  of  Pennsylvania. 


te  fcg  tje  ^euiuc  Utagram. 


ONE    HUNDRED    AND    NINE    ILLUSTRATIONS, 


SECOND  EDJTION,   REVISED  AND   ENLARGED. 
FOURTH    THOUSAND. 


NEW  YORK : 
JOHN    WILEY    &    SONS, 

LONDON: 

CHAPMAN  &  HALL,  LTD. 
1 92  <3  «\~  ;  \; 


COPYRIGHT,  1890, 

BY 
H.  W.  SPANGLER. 


£hp  Srirnttfir  flrce* 
Drummonft  anb  (Company 


PREFACE. 


THE  writer,  needing  a  book  for  class  use  which  would 
give  in  one  volume  those  parts  of  the  theory  of  valve-gears 
necessary  to  a  clear  understanding  of  the  subject,  has  pre- 
pared the  following  work. 

All  the  standard  text-books  on  the  subject,  the  current 
periodicals,  and  working  drawings  have  been  called  on  for 
data  and  methods,  and  the  works  of  Zeuner,  Auchincloss, 
Rankine,  Whitham,  Halsey,  Marks,  Reuleaux,  Bilgram,  and 
the  files  of  Engineering  and  the  Engineer  have  been  freely 
used  in  preparing  the  text ;  but  the  matter  has  been  put  in 
its  present  shape  by  the  author. 

A  few  of  the  methods  are  original,  but  others  confronted 
with  the  same  problems  have  probably  solved  them  in  the 
same  or  in  a  better  way. 

The  designing  of  valve-gears  is  entirely  a  drawing-board 
process  ;  and  in  all  but  radial  gears,  and  to  a  great  extent 
even  there,  the  actual  method  of  laying  down  the  work  is 
given. 

The  mathematical  proof  of  the  methods  and  results  used 
is  given  whenever  possible. 

The  problems  are  in  most  cases  made  up  from  the  data 
of  engines  actually  in  use. 

H.  W.  SPANGLER. 

UNIVERSITY  OF  PENNSYLVANIA, 

PHILADELPHIA,  PA.,  August  20,  1890. 

iii 


CONTENTS. 


CHAPTER   I. 

PLAIN   SLIDE  VALVES. 

1.  Plain  sliue-valves,          ..........  r 

2.  Method  of  action  of  valve,    .........  2 

3.  The  eccentric, 2 

4.  Valve  seat,  face,  and  ports,  .........  3 

5-  Lap, 4 

6.  To  determine  position  of  valve  and  piston,           .....  4 

7.  Distance  valve  has  moved  from  its  central  position,     ....  5 

8.  Yoke-connection, 5 

9.  Vaive-diagrams, 6 

10.  Angle  between  crank  and  eccentric  90°, 7 

CHAPTER   II. 

THE   ZEUNER   DIAGRAM. 

11.  To  draw  the  valve -diagram, 10 

12.  Point  of  admission,       ..........  10 

13.  Angular  advance, n 

14.  Lead,    .............  12 

15.  From  a  given  engine  to  draw  the  diagram,           .         .         ,         .         .  13 

1 6.  Distribution  of  steam  as  shown  from  the  diagram,        ....  14 

17.  Separate  diagrams  for  each  end  of  the  cylinder,  .....  14 

CHAPTER   III. 

OVERTRAVEL  AND  PROBLEMS. 

18.  Overtravel, 19 

19.  Problem  i,  given  r,  d,  cut-off  and  exhaust  closure,       ....  19 
«3.   Problem  2,  given  lap,  exhaust  lap,  lead  and  cut-off,     .         .         .         .20 

v 


VI  CONTENTS. 

21.  Problem  3,  given  cut-off,  angle  of  lead,  port  and  overtravel,         ,  21 

22.  Problem  4,  given  cut-off,  lead  and  port-opening,         •.        v        .  22 

23.  When  the  piston  and  eccentric  rods  do  not  travel  on  parallel  lines,  .         23 

24.  To  determine  the  position  of  the  eccentric,           .         .•       ,        .  «         24 

25.  Effect  of  changing  dimensions,     .         .        .                 .        •        ,  .        25 


CHAPTER    IV. 

MODIFICATIONS  OF  THE  PLAIN  SLIDE-VALVE. 

26.  Double-ported  valves,  .        .        .        .        .        .         .        .        .28 

27.  Allen  or  Trick  valve, 28 

28.  Piston-valves, 30 

29.  Taking  steam  inside, 30 

30.  Two  or  more  valves, 31 

CHAPTER  V. 

EQUALIZING  CUT-OFF,   LEAD,    COMPRESSION,   AND  RELEASE. 

31.  Equalizing  cut-off,         .                 34. 

32.  Equalizing  cut-off  and  lead,           .         .         .         .         .         .        %        .  35 

33.  Equalizing  exhaust  and  compression, 36 

34.  Circular  diagram  for  determining  movement  of  piston,       .        .        .  37 

CHAPTER  VI. 

DESIGNING  AND   SETTING  VALVES. 

35.  Designing  a  plain  slide-valve, 40 

36.  To  determine  approximate  solution,     .......  41 

37.  Equalizing  lever,           ...                  ......  43 

38.  To  put  the  engine  on  the  centre,           .......  45 

39.  To  set  the  valve 45 

CHAPTER  VII. 

THE  STEPHENSON  LINK. 

40.  The  link, 47 

41.  Point  of  suspension, 48 

42.  Slip  of  block 48 

43.  Radius  of  the  link,        ....                          ....  50 

44.  Kinds  of  links, 50 


CONTENTS.  Vit 
CHAPTER  VIII. 

THE  VALVE-DIAGRAM. 

45.  Travel  of  the  valve, .  52 

46.  The  valve-diagram, 55 

47.  Curve  of  centres, c6 

48.  To  lay  down  the  valve-diagram,  .         .         .         .         .         .         .         .56 

49.  The  virtual  eccentric, ^g 

50.  Designing  the  gear, 58 

51.  Valve-stem  and  eccentric-rod, 58 

52.  Length  of  link, .50 

53.  The  hanger, 59, 

54.  Link  suspended  at  bottom  or  centre  of  chord, 60 

55.  Open  and  crossed  rods,         . .61 

CHAPTER  IX. 

EQUALIZING  LEAD  AND  CUT-OFF. 

56.  Equalizing  lead, ,        .  63: 

57.  Equalizing  cut-off,         ..........  65 

58.  To  lay  down  the  motion,      .........  67 

59.  To  lay  down  the  centre  of  the  travel  of  the  valve,         ....  67 

60.  To  determine  the  centre  of  suspension  of  the  hanger,  ....  68 

61.  Position  of  stud,            .                                             68 

62.  Reducing  slip, .  70 

63.  Error  of  the  Zeuner  diagram, 70 

CHAPTER  X. 

THE  GOOCH  MOTION. 

64.  The  Gooch  link,  .        . 75, 

65.  Movement  of  the  valve, 76 

66.  Constant  lead, 78 

67.  Radius  of  link, 78 

68.  Suspension-rod, 79 

69.  The  hanger,           ...........  80 

70.  The  valve-diagram, 80 

71.  To  design  a  Gooch  motion,           ....•*•»  82 

CHAPTER  XI. 

THE  ALLEN  AND   FINK   MOTIONS. 

72.  The  Allen  link-motion, 85 

73.  The  valve-diagram, •        ••••8s 


Vlll  CONTENTS. 

74.  The  Fink  motion,         .         .         .        ......         .  87 

75.  Radius  of  link,      .         .         .    *    .         .         .         .         .         .         .         .  87 

76.  Suspension  of  link,        .         .         .         .         .         .-       .         .         ,         .  89 

77.  Movement  of  the  valve,        .         .         .         .         .         •        .         .         .  89 

78.  The  valve-diagram,       .         .         .         .         .                  .        ...  91 

79.  Radrus-rod  at  a  fixed  point  in  the  link, 92 

80.  Hanger  for  radius-rod,          .........  92 

Si.  Setting  the  eccentric,    .         .         .         .         .         .         .                  •.  93 

82.   Designing,    ............  94 

33.  The  Porter- Allen  motion, .".  94 

CHAPTER  XII. 

SHAFT    REGULATION. 

£4.  Throttling  governors, 98 

55.  Changing  angular  advance,  .........  98 

56.  Changing  the  eccentricity,    .........  99 

87.  Changing  eccentricity  and  angular  advance, 99 

88.  Erie  governor,     ...........  100 

Sg.  Armington  and  Sims, .      .  101 

•90.   Ball, 104 

<)i.  The  valve,     ............  105 

CHAPTER  XIII. 

RADIAL   GEARS — HACKWORTH'S. 

92.  Radial  gears,       .....                  .....  107 

93.  Hackworth's  gear,      ..........  107 

94.  Constant  lead,     ...........  108 

95.  Movement  of  the  valve, 108 

96.  The  valve -diagram, 109 

•97.  To  design  the  gear, no 

98.  Right-hand  rotation,    .  ........in 

99.  Errors  of  the  diagram,        . in 

TOO.   Port-opening,      .         ..........  113 

lor.  Connecting  up  a  Hackworth  gear, 113 

102.  Attaching  valve-stem  outside,     .         .         .         .         .         .         .         .113 

103.  Equalizing  port-opening, 113 

104.  Equalizing  the  cut-off,         .         .         .         .         .         .         .         .         .115 

CHAPTER   XIV. 

RADIAL    GEARS — MARSHALL,    ANGSTROM,    AND  JOY. 

05.   Marshall's  gear,           ..........  117 

106.   Errors  of  the  Zeuner  diagram,    .         .         .         ..         .         .         .117 


CONTENTS.  IX 

107.  Proportions  of  the  gear,     .         .         .         .         .         .         .         .         .119 

108.  Designing,  ...........  119 

109.  Angstrom's  gear,         .         .         .         .         .         .         .         .         .         .120 

no.  The  diagrams,     ...........  121 

in.   Advantage  and  disadvantage  of  radial  gears.      .....  121 

112.  The  Joy  gear, 121 

113.  Movement  of  the  valve,       .........  123 

114.  Errors  of  the  Zeuner  diagram,    ........  124 

CHAPTER   XV. 

DOUBLE  VALVES— GRIDIRON  VALVE. 

115.  Kinds  of  double  valves,       .........  127 

116.  Gridiron  valve,   ...........  127 

117.  Polonceau  valve,         ..........  128 

118.  Diagram  for  gridiron  valve,        ........  129 

119.  Combined  diagrams  for  both  eccentrics,     .         .         .         .         .         .130 

120.  Limits  of  cut-off,         ..........  131 

121.  Width  of  ports,  ...........  132 

122.  Angle  of  advance,        ..........  132 

123.  Varying  cut-off,  .         . 133 

124.  Arrangement  used  for  varying  cut-off,         ......  134 

125.  Width  of  cut-off  valve,         .........  134 

126.  Varying  width  of  block, .         .        .135 

CHAPTER   XVI. 

RELATIVE  MOVEMENT— POLONCEAU   GEAR. 

127.  One  valve  on  the  back  of  another,       .         .         .         .         .         .         .  137 

128    Relative  valve-circle,  ..........  138 

129.  To  draw  the  relative  valve-circle,        .......  138 

130.  The  Polonceau  valve,          .........  139 

131.  The  Polonceau  gear,  ..........  139 

132.  Valve-diagram,   ...........  139 

133.  Limits  of  cut-off,         ..........  140 

134.  Dimensions  of  valve,  .........  141 

CHAPTER   XVII. 

BUCKEYE   GEAR. 

135.  The  valve, 143 

136.  The  eccentrics  and  connections,  .         .         .         .         .          .         .  143 

137.  Movement  of  the  valves,     .........  144 


X  CONTENTS. 

138.  Cut-off  valve-diagram,         ....        3"'.  f-       .         .         .  145 

139.  Changing  cut-off, "      ,  .      «         «    -.  •_  ;^         .  145 

140.  The  governor,     .        .        .        .        .        .        •„».*«•.  146 

CHAPTER   XVIII. 

MEYER  VALVE  AND  GUINOTTE  GEAR. 

141.  The  Meyer  valve,        .         .         .         ...        .        .        .        *        .148 

142.  Changing  the  distance  between  the  blocks,          .....  148 

143.  Designing  a  Meyer  valve, 150 

144.  Length  of  cut-off  blocks .151 

145.  Cut-off  with  inside  edges,    .         .         .         .         .        .         .         .         .151 

146.  Guinotte's  gear,           .         .         .         .         .         .         .         .'»'•*-  152 

147.  Movement  of  the  valve,       .         .         .         .         .         .                  .         «  153 

148.  To  draw  the  valve  diagram, .        .154 

CHAPTER  XIX. 

BILGRAM,  REULEAUX,   AND  ELLIPTICAL   DIAGRAMS. 

149.  Bilgram  diagram, 157 

150.  Problems,           .         .         .         .         .         .         .         .         •        »        .  158 

151.  Reuleaux's  diagram,  .                  160- 

152.  Problems "  »        •        .        .  161 

153.  Elliptical  diagrams,              165 

154.  Velocity  of  the  valve, 164 

CHAPTER   XX. 

CORLISS     VALVE-GEAR. 

155.  Hamilton-Corliss  engine, 166 

156.  Movement  of  the  valve,     . .  168- 

357.   Proportioning  the  parts,    .......                  .  169 


SYMBOLS. 


A.  Aoscissa  of  the  end  of  that  diameter  of  the  valve-diagram 

which  passes  through  the  origin. 

B.  Ordmate  of  the  end  of  that  diameter  of  the  valve-diagram 

which  passes  through  the  origin. 
L.  Distance  from  outside  to  outside  of  ports  in   top  of  Meyer 

valve. 
./?.  Radius  of  crank. 

a.  Length  of  eccentric-rod  in  Fink  gear  to  attachment  of  sus- 

pension-rod. 

b.  Length  of  connecting-rod. 

Length  of  eccentric-rod  in  Fink  gear  from  attachment  of 
suspension-rod  to  link.  .. 

c.  One  half  the  chord  of  the  link. 

Distance  from  cross-head  to  point  in  connecting-rod  at  which 

radius-rod  in  Joy  gear  is  attached. 
t.  Width  of  port  in  upper  valve-seat  of  double  valves, 
y.  Width  of  port  in  upper  valve  of  double  valves. 
g.  Length  of  eccentric-rods. 
gv  Length  of  radius-rod  in  link-motions. 
g-2.  Length  of  valve-stem. 
h.  Length  of  hanger  in  link-motions. 
/.  Lead. 
/.  Lap. 
/j.  Length    of   radius-rod   from   closed   curve   to  open   one  in 

radial  gears. 
/a.   Distance  from  end  of  valve  connecting-rod  to  open  curve  in 

radial  gears. 

/3.  Distance   from    point   on   connecting-rod   to  point  on  open 
curve  in  the  Joy  gear. 

xi 


xii  SYMBOLS. 

/4.  Distance  from  point  on  connecting-rod  to  point  of  attach- 
ment of  secondary  radius-rod  in  Joy  gear. 

n.   Revolutions  of  the  engine  per  minute. 

/.   Port-opening. 

r.   Radius  of  eccentric  or  eccentricity. 

fj.  Throw  of  eccentric  moving  second  or  cut-off  valve;  and  in 
link-motions,  where  unequal  eccentricities  are  used,  of  the 
second  one. 

rx.  Diameter  of  the  relative  valve-circle  with  double  valves. 

s.  Distance  the  cut-off  valve  has  to  move  from  its  central  posi- 
tion to  close  the  port. 

u.  Distance  between  the  centre  of  the  link  and  the  end  of  the 
valve-stem  or  radius-rod  in  Stephenson,  Gooch,  and  Fink 
links/  In  Allen  motion  distance  end  of  radius-rod  has 
moved  from  its  central  position. 

«,.  In  Allen  motion,  distance  the  link  has  moved  from  its  cen- 
tral position. 

x.  Distance  the  valve  has  moved  from  its  central  position  for 
any  position  of  the  crank. 

y.  Half  the  distance  between  the  blocks  in  a  Meyer  gear. 

of.  Fixed  angle  between  the  crank  and  the  eccentric,  angle  be- 
tween eccentric-rod  and  the  centre  line  of  its  motion  in  a 
Fink  motion,  and  angle  of  path  of  end  of  radius-rod  in 
radial  gears. 

0.  Ninety  degrees  less  than  the  angle  between  the  dead-point 
and  point  of  cut-off. 

y.  |  Angles  used  in  finding  movement  of  the  valve  in  link-mo- 
y'.  \      tions. 

6.  Angular  advance  of  main  eccentric. 

<£j.  Angular  advance  of  cut-off  eccentric  or  when  different  angles 
are  used  in  link-motions  of  the  second  one. 

tfa.  In  Guinotte  gear  angular  advance  of  third  eccentric. 

8.  Angle  described  by  hanger  in  link-motions. 
Angle  used  in  Joy  gear  demonstration. 

0.  Angle  used  in  Joy  gear  demonstration. 

Auxiliary  angle  used  in  proving  the  construction  of   Prob- 
lem 4,  page  22. 

<r.  Angle  crank  is  moved  in  a  Stephenson  link  to  equalize  lead. 

co.  Angle  the  crank  has  moved  from  its  central  position 


VALVE-GEARS. 


CHAPTER  I. 

PLAIN  SLIDE-VALVES. 

1.  Plain  Slide-Valves.— In  an  ordinary  steam-engine 
steam  is  admitted  to  and  released  from  each  end  of  the  steam- 
cylinder  by  a  valve  actuated  by  the  engine  itself.  As  the 
economical  working  of  an  engine  depends  to  a  very  great 
extent  upon  the  proper  admission  and  release  of  the  steam, 
a  study  of  the  valves  used  and  of  the  methods  of  moving 
them  is  important. 

Fig.  i  is  a  sketch  of  the  valve  ordinarily  used,  which  is 
often  called  a  D  slide-valve  from  its  shape  and  method  of 


FIG.  i. 

action,  a  is  the  valve,  b  is  the  passage  leading  to  one  end 
of  the  cylinder,  and  c  that  leading  to  the  other.  These  are 
called  the  steam-passages,  d  is  a  passage  leading  to  the 
open  air  or  to  a  condenser,  and  is  called  the  exhaust-passage. 
The  space  e,  or  valve-chest,  is  filled  with  steam,  and  is  in 
direct  communication  with  the  boiler. 


2.  Method  of  Action  of  Valve.— The  action  of  the  valve 
is  as  follows:  Suppose  the  valve  a  is  moved  to  the  right. 
Steam  passes  from  the  space  e  through  c  to  the  left-hand  end 
of  the  cylinder,  and  moves  the  piston  to  the  right.     Any 
steam  that  may  be.  in  the  right-hand  end  of   the  cylinder 
passes  through  the  passage  b  into  the  space  f  under  the 
valve,  and  thence  through  d  away.     When  the  piston  has 
reached  the  end  of  its  stroke  to  the  right,  the  valve  a  has 
moved  back  far  enough  towards  the  left  to  allow  steam  from 
^  to  pass  into  b,  and  the  passage  c  is  connected  with  /,  thus 
causing  the  piston  to  move  towards  the  left. 

3.  The  Eccentric. — The  mechanism  connecting  the  valve 
-and  piston  in  its  simplest  form  is  shown  in  Fig.  2.     The  rod 


FIG.  2. 

ge  is  connected  to  the  valve  and  is  called  the  valve-stem. 
The  rod  ec  connects  the  end  of  the  valve-stem  with  the  crank 
ac,  turning  around  the  point  a,  which  is  the  centre  of  the 
shaft.  The  rod  ec  is  called  the  eccentric-rod,  fd  is  a  rod 
connected  at  one  end  to  the  piston,  and  is  called  the  piston- 
rod,  db  is  a  rod  connecting  the  end  d  of  the  piston-rod  with 
the  end  b  of  a  crank  ab,  which  is  rigidly  connected  to  ac,  and 
turns  with  it  about  a.  db  is  called  the  connecting-rod.  The 
crank  ab  will  be  spoken  of  hereafter  as  "  the  crank,"  and  the 
•crank  ac,  which  is  the  mechanical  equivalent  of  the  eccentric, 
will  be  spoken  of  as  "  the  eccentric."  The  rods  ge  and  fd  are 
supposed  to  move  along  the  line  ah,  but  are  separated  in 
the  figure  for  clearness. 

The  arrangement  actually  used  is  shown  in  Fig.  3.  The 
inner  circle  with  a  as  a  centre  is  the  shaft.  The  inner  circle 
having  c  as  a  centre  is  the  eccentric  b  or  the  eccentric 


PLAIN  SLIDE-VALVES. 


sheave,  and  is  keyed  to  the  shaft  and  turns  with  it.  The 
outer  broken  circle  having  c  as  a  centre  is  the  eccentric 
strap /f,  which  turns  easily  on  the  sheave  c,  but  is  rigidly 


FIG.  3. 

attached  to  the  bar  d.  If  the  shaft  turns,  the  movement  of 
point  e  along  the  line  ae  would  be  exactly  the  same  with  this 
arrangement  as  though  ac  were  a  crank  and  ec  a  rod,  and 
the  representation  in  Fig.  2  is  practically  the  same  as  in 
Fig.  3.  The  distance  ac  is  called  the  eccentricity. 

4.  Valve-Seat,  Face,  and  Ports. — Referring  again  to 
Fig.  I,  that  part  on  which  the  valve  moves  is  called  the 
valve-seat.  That  part  of  the  valve  sliding  over  the  seat  is 
called  the  valve-face.  The  openings  through  the  valve-seat 


FIG.  4. 

to  the  passages  b,  c,  and  d  are  called  the  ports,  those  lead- 
ing to  b  and  c  being  the  steam-ports,  and  to  d  the  exhaust- 
port.  A  plan  of  the  valve-seat  is  shown  in  Fig.  4,  in  which 
c  and  b  are  the  steam-ports,  and  d  the  exhaust-port. 


VALVE-GEARS. 


5.  Lap. — In  Fig.  I  the  valve  is  shown  as  covering  both 
ports  equally,  and  is  said  to  be  in  its  middle  position.  Fig. 
5  shows  one  end  of  the  valve  to  a  larger  scale.  The  distance 


FIG.  5. 

ik  that  the  valve  overlaps  the  outside  of  the  steam-port  when 
in  mid  position  is  called  the  steam  or  outside  lap,  or  simply 
the  lap.  The  distance  kl  that  the  valve  overlaps  the  inside 
edge  of  the  steam-port  is  called  the  inside  or  exhaust-lap. 
It  is  evident  that  before  the  passage  b  can  receive  steam  the 
valve  must  move  to  the  left  a  distance  ih,  or  the  lap ;  and 
before  b  can  be  open  to  exhaust,  the  valve  must  move  to  the 
right  a  distance  kl,  or  the  exhaust-lap,  from  its  middle  posi- 
tion. 

6.  To  Determine  Position  of  Valve  and  Piston. — In 
Fig.  6  a  is  the  centre  of  the  shaft,  ab  one  position  of  the 


FIG.  6. 


crank,  and  ac  the  corresponding  position  of  the  eccentric. 
If  the  length  of  the  connecting-rod  bd,  the  length  of  the 
piston-rod  df,  and  the  direction  af  of  the  piston  travel  are 
known,  the  position  of  the  piston  corresponding  to  the  posi- 


PLAIN  SLIDE-VALVES. 


5 


tion  ab  of  the  crank  can  be  determined  by  laying  off  irom 
b  a  distance  bd  equal  to  the  length  of  the  connecting-rod, 
thus  determining  the  position  of  d,  and  from  d  laying  off  the 
length  of  the  piston-rod  to/",  thus  determining  the  position 
of  the  piston.  Similarly,  if  the  length  of  the  eccentric-rod 
ce  and  of  the  valve-stem  eg  are  known,  the  position  of  the 
valve  can  be  determined. 

7.  Distance  Valve  has  moved  from  its  Central  Position. 
— In  most  engines  ec  is  very  long  as  compared  with  ac. 
When  ac  is  vertical,  as  at  ac' ,  the  valve  is  practically  in  its 
middle  position,  and  the  distance  it  is  from  its  middle  posi- 
tion when  the  eccentric  occupies  any  position,  as  ac,  can  be 
represented  by  ah,  the  line  ch  being  perpendicular  to  hf.  In 


FIG.  7. 

the  case  of  the  piston  this  is  not  so,  as  the  rod  bd  is  gener- 
ally four  to  six  times  ab,  and  the  position  of  the  piston 
materially  depends  on  bd.  However,  if  the  point  b  can 
always  be  determined,  the  position  of  the  piston  can  also. 

8.  Yoke  Connection. — Fig.  7  is  a  case  in  which  both  valve- 
stem  and  piston-rod  are  connected  to  slotted  crossheads, 
the  connections  being  known  as  yoke  connections.  The  dis- 


0  VALVE-GEARS. 

tance  the  piston  has  moved  irom  its  central  position  is  ak, 
and  the  distance  that  the  valve  has  moved  is  aL  In  the 
figure  the  piston-rod  /  has  attached  to  its  end  the  slotted 
piece  d,  in  which  the  block  b  attached  to  the  crank  moves. 
The  end  of  the  valve-stem  g  has  a  similar  piece  e  attached, 
in  which  the  block  c  attached  to  the  eccentric  ac  moves. 

To  show  that  ac  is  the  right  position  for  the  eccentric 
corresponding  to  the  position  ab  of  the  crank,  if  the  engine 
is  to  turn  in  the  direction  of  the  arrow,  let  ah  be  one  dead 
point,  that  is,  one  position  where  the  crank  and  connecting- 
rod  are  in  the  same  straight  line.  As  the  piston  is  now  to 
move  to  the  right,  the  valve  must  have  already  moved  to 
the  right  a  sufficient  distance  to  admit  steam  to  the  left- 
hand  end  of  the  cylinder,  and  should  be  moving  in  the  direc- 
tion that  would  open  the  port  still  wider.  This  could 
occur  only  if  the  eccentric  occupied  the  position  indicated 
by  ok,  and  could  not  occur  if  the  eccentric  was  at  ai.  The 
angle  kah  is  therefore  the  fixed  angle  between  the  crank  and 
the  eccentric,  and  when  the  crank  reaches  ab  the  eccentric 
is  at  ac.  This  combination  is  usually  spoken  of  as  a  valve 
with  an  infinite  eccentric-rod,  and  a  piston  with  an  infinite 
connecting-rod. 

9.  Valve -Diagrams. — Valve-diagrams  are  used  to  show 
at  a  glance  the  movement  of  the  valve  for  any  movement  of 
the  piston,  and  the  various  events  occurring  in  a  stroke  of 
the  piston.  Numerous  forms  of  diagrams  are  used,  all  more 
or  less  accurate  and  convenient,  the  form  used  throughout 
this  work  being  that  proposed  and  developed  by  Dr.  Gus- 
tav  Zeuner  in  his  admirable  treatise  on  valve-gears,  as  it  is 
by  far  the  most  convenient  to  use  of  any  that  have  been  pre- 
pared, and  is  as  accurate  as  any  of  them. 

The  diagram  for  the  case  of  the  slotted  connections  above 
described  will  first  be  determined,  and  if  any  ordinary  con- 
necting-rod,  as  shown  in  Fig.  6,  be  used  instead  of  that 
shown  in  Fig.  7,  the  position  of  the  piston  corresponding  to 
any  position  of  the  crank  can  readily  be  found  as  shown  in 


PLAIN  SLIDE-VALVES.  7 

Fig.  6.     For  Fig.  7  the  diagram  to  be  determined  is  exactly 
correct. 

Suppose  the  crank  to  start  from  the  position  ah  of  Fig.  7, 
and  to  move  through  the  angle  GO  to  the  position  ab.  Call 
the  fixed  angle  between  the  crank  and  the  eccentric  or. 
Then  the  distance  the  valve  is  from  its  central  position  is 
al  =  ac  cos  cal  •=.  ac  cos  (180  —  ex.  —  GO)  =  —  ac  cos  (a  -j-  G?), 
or  calling  ac  =  r  and  al  =  x, 

x  —  —  r  cos  (or  +  GO) (l) 

If  when  the  crank  is  just  on  its  dead  point  the  valve  is 
just  in  its  middle  position,  the  angle  a  =  90,  and 


x  =  —  r  cos  (90  +  GO)  =  r  sin  CD  ;    . 


(2) 


and  this  case  we  will  examine  first. 

10.  Angle  between  Crank  and  Eccentric  90°. — In  Fig. 
8  let  ah  represent  the  position  of  the  crank  at  one  dead- 


\ 


point,  and  suppose  the  crank  turns  around  a  against  the 
hands  of  a  clock.  Let  ab  be  any  other  position  of  the  crank 
after  it  has  moved  from  the  dead-point  the  angle  GO.  As  we 
have  already  seen,  the  valve  has  moved  from  its  central  posi- 
tion a  distance  x  —  r  sin  GO. 


8  VALVE-GEARS. 

On  the  line  ab  lay  off  the  distance  ac  =  r  sin  GJ,  and  for 
every  other  position  of  the  crank  lay  off  the  corresponding 
value  of  x.  The  result  will  be  a  series  of  points,  the  curve 
passing  through  which  will  be  a  circle  whose  diameter  is  r, 
and  whose  centre  is  on  ad.  For  if  we  lay  off  on  ad  a  dis- 
tance r,  and  draw  a  circle  on  ad  as  a  diameter,  then  if  c  is 
any  point  on  the  circumference,  the  angle  dca  is  a  right 
angle,  and 

ac  =  r  cos  dac  —  r  cos  (90  —  GO)  =  r  sin  GO  =  x. 

QUESTIONS. 

1.  Draw  a  plain  D  slide-valve  in  its  middle  position,  and 
name  all  the  parts. 

2.  How  must  the  valve  move  to  cause  the  engine  to  run  ? 

3.  Sketch  the  arrangement  by  which  the  valve  is  moved. 

4.  What  is  meant  by  lap  ? 

5.  If  the  valve  took  steam  inside  and  exhausted  outside, 
what  would  then  be  the  steam-lap  ? 

6.  In  a  given  engine,  how  determine  the  actual  position 
of  the  valve  corresponding  to  a  given  piston  or  crank  posi- 
tion? 

7.  Why  can  the  length  of  the  eccentric-rod  be  neglected 
in  this  work  ? 

8.  What  is  the  objection  to  neglecting  the  length  of  the 
connecting-rod  in  determining  the  distance  the  piston  has 
travelled  for  any  movement  of  the  crank? 

9.  What  should  be  the  relative  position  of  the  crank  and 
eccentric  to  turn  in  any  given  direction,  the  connections 
being  as  shown  in  Fig.  6  ? 

10.  What  is  meant  by   an   infinite  connecting-rod,  and 
what  is  its  equivalent  as  far  as  the  movement  of  the  piston  is 
concerned  ? 

11.  What  is  the  use  of  valve-diagrams  ? 

12.  For  any  given  movement  GO  of  the  crank  from  its 
dead-point,  how  far  has  the  valve  moved  from  its  central 
position? 


PLAIN  SLIDE-VALVES. 


13.  How  would  you  construct  a  valve-diagram,  having 
given  the  distance  the  valve  has  moved  from  its  central  posi- 
tion for  various  values  of  cap 


PROBLEMS. 

1.  Given  the  crank  10  inches,  connecting-rod  50  inches, 
and  suppose  the  crank  to   be  on  that  dead-point  farthest 
from  the  cylinder,  how  far  has  the  piston  moved  for  each 
30°  of  its  revolution  ? 

2.  In  a  parallel  column,  put  down  the  distance  the  piston 
would  have  moved  had  the  piston  and  crank  been  connected 
by  a  yoke. 

3.  Given  the  eccentricity  equal  to  3  inches,  and  the  angle 
between  the  crank  and  eccentric  equal  to  135°,  calculate  the 
distance  the  valve  has  moved  for  each  30°  of  movement  of 
the  crank,  plot  the  points,  and  draw  the  valve-diagram. 


CHAPTER   II. 
THE  ZEUNER  DIAGRAM. 

11.  To   draw  the  Valve-  Diagram.  —  Instead,  therefore, 
of  laying  down  the  points  separately,  if  on  ad,  Fig.  8,  we 
lay  off  r  and  draw  a  circle  on  r.  as  a  diameter,  we  can  deter- 
mine the  distance  the  valve  has  travelled  from  its  middle 
position  for  any  movement  of  the  crank  by  drawing  the 
position  of  the  crank;  and  that  part  of  the  crank  line  lying 
between  a  and  the  circumference  of  the  circle  is  the  move- 
ment of  the  valve. 

When  the  crank  has  reached  any  point  below  the  hori- 
zontal line  as  ae,  the  distance  the  valve  has  moved  is  af,  and 
is  measured  in  the  opposite  direction.  Evidently,  from  the 
diagram,  when  the  crank  is  on  either  dead-point,  the  valve 
is  just  at  its  central  position,  as  we  assumed  to  begin  with. 

12.  Point  of  Admission.  —  From  an  inspection  of  Fig.  i, 
it  is  seen  that  if  the  crank  is  on  the  dead-point  when  the 
valve  is  in  its  central  position  and  covers  both   ports,  no 
steam  could  be  admitted  to  the  cylinder,  and  the  engine 
would  have  no  tendency  to  start. 

We  can  determine  from  the  diagram  when  steam  will  be 
admitted.  From  Fig.  5  it  is  evident  that  the  distance  the 
port  into  b  is  opened  when  the  valve  has  moved  a  distance 
x  from  its  middle  position,  is  x  —  ih  =  x  —  the  lap,  and  call- 
ing/ the  port-opening  and  /the  lap, 


(3) 


That  is,  in  Fig.  8,  if  from  the  distances  ac,  ad,  af,  etc.,  we  take 
a  distance  equal  to  the  lap,  the  portion  remaining  is  the 
opening  of  the  port.  With  a  as  a  centre  and  a  radius  al  = 


THE   ZbUNER   DIAGRAM.  IT 

the  lap,  draw  the  circle  hklm,  called  the  lap-circle.  Then 
when  the  crank  reaches  ab  the  opening  of  the  port  is  the 
distance  Ic,  and  similarly  for  any  other  position  of  the  crank. 
At  ak  the  lap  just  equals  the  movement  of  the  valve,  and  the 
port  is  just  about  to  open.  When  the  crank  reaches  am  the 
movement  of  the  valve  from  its  central  position  is  again 
equal  to  the  lap,  and  the  valve  has  just  closed  the  port  to 
steam. 

13.  Angular  Advance. — In  order  that  the  full  pressure  of 
the  steam  may  come  on  the  piston  at  the  beginning  of  the 
stroke,  the  angle  a  between  the  crank  and  eccentric  is  never 
made  equal  to  90°,  but  something  greater.  The  increase  of 
the  angle  a  is  called  the  angular  advance. 

The  angular  advance  may  be  defined  as  the  angle  be 
tween  the  actual  position  of  the  eccentric  and  that  position 
of  the  eccentric  which  would  bring  the  valve  to  its  middle 
position,  the  crank  being  in  both  cases  at  a  dead-point. 

This  angle  of  advance  we  will  call  d,  and  a  =  90  -f-  d. 
We  have  then,  from  equation  (i), 

x  =  —  r  cos  (90  +  $  +  <*>)  =  r  sin  (<*  +<»)••     •     (4) 

Evidently,  if  GO  =  90  —  6,  x  =  r. 

In  Fig.  9  the  same  diagram  has  been  drawn  as  in  Fig.  8r 
but  the  diameter  of  the  valve-circle  has  been  moved  through 
an  angle  bad  =  <5. 

O 

In  this  figure,  if  ac  =  x  =  r  sin  (tf+  GO),  the  circle  still 
represents  the  valve-diagram.  For  ad  =  r  and  dca  is  a  right 
angle; 

.*.  ac  —  ad  cos  doc  =  r  cos  (90  —  d  —  GO)  —  r  sin  ($  -{-  GO)  =  x. 

It  is  to  be  remembered  that  the  valve-diagram  shows  the 
movement  of  the  valve  for  varying  positions  of  the  crank, 
and  the  centre  line  of  the  valve-diagram  as  ad  in  Fig.  9  is 
not  the  position  of  the  eccentric  when  the  crank  is  at  an. 
The  small  diagram  shows  the  relative  position  of  the  crank 
and  eccentric  to  give  the  valve-diagram  shown  in  the  figure. 


12 


VALVE-GEARS. 


We  see  that  by  giving  the  eccentric  angular  advance  we 
have  simply  moved  the  valve-circle  about  a  through  the 
angle  6.  It  will  be  seen  that  the  port  now  opens  when  the 


FIG.  9. 

crank  is  at  ak,  and  therefore  steam-pressure  is  on  the  steam- 
piston  at  the  beginning  of  the  stroke.  The  angle  hak  is 
often  called  the  angle  of  lead. 

14.  Lead. — When  the  crank  is  on  the  dead-point  an,  the 
distance  hn  is  evidently  the  opening  of  the  port.  This  dis- 
tance is  called  the  lead.  The  lead  can  therefore  be  defined 
as  the  amount  the  oort  is  open  to  steam  at  the  beginning  ot 


8 — *--4 


FIG.  10. 


the  stroke,  or  when  the  crank  is  on  its  dead-point.  In  the 
same  way  exhaust-lead  is  the  amount  the  steam-port  is  open 
to  exhaust  at  the  beginning  of  the  stroke. 


THE   ZEUNER  DIAGRAM. 


15.  From  a  given  Engine  to  draw  the  Diagram. — Hav- 
ing any  valve  given  moved  by  a  single  eccentric,  we  can 
lay  down  the  valve-diagram  and  determine  the  points  of 
admission  and  cut-off,  the  opening  and  closing  of  the  ex- 
haust, etc. 

In  Fig.  10,  suppose  the  valve  to  be  a  inches  over  all  and 
b  inches  inside,  the  steam-port  to  be  c  inches  wide,  the  ex- 
haust-port d  inches  and  the  bridges  or  material  between  the 
ports  to  be  e,  and  the  eccentricity  to  be  r  inches,  and  the 


angular  advance  d  degrees,  to  determine  the  various  points 
relating  to  the  valve. 


From  the  figure  the  lap  = 
2e  +  d  -  b 


a  —  d —  2e 


2C 


and  the  ex. 


haust-lap  = 


.     To  draw  the  diagram  in  Fig.    n 


14  VALVE-GEARS. 

draw  the  two  lines  ab  and  cd  at  right  angles  to  each  other. 
Let  od  be  one  dead-point,  the  engine  to  turn  in  the  direction 
of  the  arrow.  Lay  off  the  line  oe  so  that  the  angle  aoe  is  d 
degrees.  On  oe  lay  off  of=  r  the  eccentricity,  and  on  of  as 
a  diameter  draw  the  valve-circle.  With  o  as  a  centre  and 
og  as  a  radius  equal  to  the  lap,  draw  the  lap-circle  gqk,  and 
with  a  radius  oh  equal  to  the  exhaust-lap  draw  the  exhaust- 
lap  circle  /is I. 

16.  Distribution  of  Steam  as  shown  from  the  Diagram. 
—When  the  crank  is  at  od,  the  steam-port  on,  say,  the  right 

side  is  open  the  distance  mp  or  the  lead,  while  the  exhaust- 
port  on  the  opposite  side,  say  left,  is  open  np  or  the  exhaust- 
lead.  When  the  crank  reaches  oe  both  ports  are  opened 
widest,  that  on  the  right  to  steam  and  on  the  left  to  exhaust. 
When  the  crank  reaches  ok  the  steam-port  on  .the  right 
closes  and  cut-off  takes  place.  When  the  crank  reaches  ol 
the  exhaust  closes  on  the  left  side  of  the  piston  and  com- 
pression begins.  When  the  crank  reaches  oh'  (oh  prolonged) 
the  exhaust  opens  on  the  right  side  of  the  piston.  At  og' 
(og  prolonged)  the  steam-port  opens  on  the  left.  At  oc  the 
steam-port  is  open  on  the  left  a  distance/?//,  and  the  exhaust 
is  open  on  the  right  a  distance  pn.  When  the  crank 
reaches  ok'  the  steam  is  cut  off  on  the  left-hand  side,  at  ol' 
the  exhaust  closes  on  the  right-hand  side.  At  oh  the  ex- 
haust on  the  left  opens  again,  and  at  og  the  steam  is  again 
admitted  on  the  right-hand  side. 

17.  Separate  Diagram  for  each  End  of  the  Cylinder. — 
To  make  this   clearer,   the   diagrams   Figs.  12  and    13    are 
drawn,  one  of  which  shows  the  distribution  of  steam  in  the 
left-hand  end  of  the  cylinder,  while  the  other  shows  the  dis- 
tribution in  the   right-hand  end.     The  letters  refer  to  the 
same  things  as  in  Fig.  1 1.    The  circles  with  oa  as  a  radius  are 
drawn  to  any  convenient  scale  to  represent  the  line  travelled 
through  by  the  crank-pin,  and  the  lines  ab  represent  the 
stroke  of  the  piston  to  the  same  scale,    gk  and  g'k'  are  the 
lap-circles,  hi  and  h'l  are  the  exhaust-lap  circles. 

In  the  right-hand  end  of  the  cylinder,  Fig.  12,  starting  at 


THF   ZEUNER  DIAGRAM. 


the  dead-point  a,  the  steam-lead  is  mp,  cut-off  takes  place  at 
2,  exhaust  opens  at  3,  and  the  port  is  open  the  distance  rip' 
or  the  exhaust-lead  at  the  beginning  of  the  return  stroke ;  at 
4  the  exhaust  closes,  and  at  i  the  steam-port  opens  again  so 


FIG.  12. 

that  when  the  crank  again  gets  to  a  the  port  is  open  the 
lead. 

On  the  lower  line  ab,  representing  the  stroke  of  the 
piston,  starting  at  a,  steam  is  admitted  until  the  piston 
reaches  6,  when  cut-off  takes  place ;  at  7  the  exhaust  opens, 
and  the  piston  travels  to  the  end  of  the  stroke.  On  the 
return  stroke,  at  8  the  exhaust-port  is  closed,  at  5  the  steam- 
port  again  opens,  and  the  piston  travels  to  the  end  of  its 
stroke  again.  Steam  is  being  admitted  while  the  crank 


16 


VALVE-GEARS. 


travels  from  i  to  2,  it  is  being-  expanded  from  2  to  3;  exhaust 
is  taking  place  from  3  to  4,  and  compression  from  4  to  i. 

While  this  is  taking  place  in  the  right-hand  end  of  the 
cylinder,  the  left-hand  end  is  also  receiving  steam  and  ex- 
hausting, as  shown  in  Fig.  13.  Starting  at  the  same  dead- 
point  a,  the  ex-haust  is  open  on  the  left  a  distance  np  equal 


FIG.  13. 

to  the  exhaust-lead  ;  at  12  the  exhaust  closes,  at  9  steam  opens 
on  the  left,  and  at  the  dead-point  b  the  port  is  open  to 
steam  a  distance  m p'  equal  to  the  steam-lead.  On  the  return 
stroke,  at  10  steam  is  cut  off,  at  1 1  exhaust  opens,  and  at  a  the 
dead-point  is  reached. 

The  line  ab  at  the  bottom  of  the  figure  shows  the  move- 
ment of  the  piston.     Moving  from  a,  at  16  exhaust  closes,  at 


THE   ZEUNER  DIAGRAM.  I/ 

13  steam  opens;  on  the  return  stroke,  at  14  cut-off  takes  place, 
and  at  15  exhaust  opens.  Exhaust  takes  place  while  the 
crank  is  moving  from  11  to  12,  compression  from  12  to  9, 
admission  from  9  to  10,  and  expansion  from  10  to  n. 

QUESTIONS. 

14.  Explain  the  method  of  drawing  the  Zeuner  diagram. 

15.  How  is  the  point  of  admission  found? 

1 6.  What  is   the   port-opening  for  any  position  of  the 
crank  ? 

17.  What  is  angular  advance,  and  why  is  it  given? 

1 8.  What  effect  has  angular    advance  on  the  valve-dia- 
gram ? 

19.  What  is  lead? 

20.  What  is  meant  by  angle  of  lead  ? 

21.  What  dimensions  are  required  to  determine  the  valve- 
diagram  for  a  given  engine? 

22.  Explain  fully  the  different  events  occurring  in  one 
end  of  the  cylinder,  in  the  order  of  their  occurrence,  during 
one  complete  revolution. 

23.  Explain  fully  the  events  occurring  in  both  ends  of 
the  cylinder,  in  the  order  of  their  occurrence,  during  one 
complete  revolution. 

24.  How  determine  the  actual  position  of  the  piston,  the 
length  of  the  connecting-rod  being  given,  for  each  event 
occurring  in  one  revolution  in  one  end  of  the  cylinder  ? 

PROBLEMS. 

4.  Given  the  eccentricity  3  inches  and  the  angle  between 
the  crank  and  the  eccentric  90°,  how  far  has  the  valve  moved 
from  its   central    position  when    09=30°,  60°,  90°?     Draw 
Zeuner  diagram  and  scale  distances. 

5.  In  the  above  problem,  if  the  lap  is  if  inches,  through 
what  angle  has  the  crank  moved  from  its  dead-point  when 
the  port  opens?     What  when  the  port  closes? 

6.  What  is  the   port-opening   when  &  =  75°,  and   what 
when  GO  =  150°? 


1 8  VALVE-GEARS. 

7.  A  given  engine  is  136  inches  from  centre  of  shaft  to 
•centre  of  exhaust-port.  The  valve  is  Q|  inches  over  all,  and 
5fJ-  inside.  The  exhaust-port  is  3  inches,  the  steam-ports  each 
if  inches,  and  the  bridges  if  inches  each,  r  =  2j".  The  length 
of  the  connecting-rod  is  90  inches,  and  the  eccentric-rod  60 
inches.  The  cylinder  is  18  inches  diameter  by  24  inches 
stroke,  and  the  angle  of  advance  is  25°.  Where  in  each 
stroke  is  steam  admitted  and  cut  off,  and  where  does  exhaust 
and  compression  in  each  end  take  place?  Measure  all  dis- 
tances for  each  end  of  the  cylinder  from  that  end  of  the 
stroke  at  which  steam  is  admitted. 


CHAPTER  III. 

OVERTRAVEL  AND  PROBLEMS. 

18.  Overtravel. — The  travel  of  the  valve  is  otten  more 
than  sufficient  to  open  the  port  wide.     The  excess  is  called 
overtravel.     Thus  in  Fig.  11,  the  distance  qr  is   the  width 
of  the  port,  and  the  port  is  wide  open  to  steam  while  the 
crank  passes  from  ox  to  oy,  and  the  distance  rf  is  the  over- 
travel. 

Similarly  st  is  the  width  of  the  port,  and  the  port  is  wide 
open  to  exhaust  from  ov  to  ow.  That  is,  on  the  right  side 
the  port  begins  to  open  at  og,  is  wide  open  at  ox,  begins  to 
close  at  oy,  and  is  entirely  closed  at  ok. 

In  the  same  way  the  exhaust  on  the  left-hand  side  begins 
to  open  at  oh,  is  wide  open  at  ov,  begins  to  close  at  ow,  and 
is  entirely  closed  at  ol.  Without  overtravel  the  valve  was 
opening  the  port  from  og  to  oe\  with  overtravel  the  port  is 
opening  while  the  crank  passes  from  og  to  ox.  Over- 
travel  therefore  causes  the  port  to  be  opened  and  closed 
more  quickly,  but  for  the  same  opening  of  the  port  the 
eccentricity  must  be  greater  by  the  amount  of  overtravel. 

The  following  are  the  ordinary  problems  that  occur  in 
designing  valves. 

19.  Problem  I. — Given  the  eccentricity  r,  the  angle  of 
advance  d,  the  point  of  cut-off,  and  the  point  of  closing  of 
the  exhaust,  to  find  the  lap,  exhaust-lap,  lead  and  exhaust- 
lead,  and  the  greatest  possible  opening  of  the  port. 

In  Fig.  14  draw  ao  and  do  at  right  angles  to  each  other. 
Lay  off  aoe  =  $,  the  angular  advance.  On  oe,  with  a  radius 

equal  to  one  half  the  eccentricity  (— J,  draw  the  valve-circle. 


20 


VALVE-GEARS. 


Draw  ok  to  represent  the  position  of  the  crank  at  cut-off, 
and  ol  for  the  closing  of  the  exhaust.  Through  the  inter- 
section  of  these  lines  with  the  valve-circle,  and  with  o  as  a 
centre,  draw  Ish  and  kqg.  Then  ol  is  the  exhaust-lap,  ok  the 


FIG.  14. 

lap,  mp  the  lead,  np  the  exhaust-lead,  qf  is  the  greatest 
possible  opening  of  tne  port  to  steam,  and  sfto  exhaust. 

20.  Problem  2. — Given  the  lap,  exhaust-lap,  point  of 
cut-off,  and  the  steam-lead,  to  determine  the  eccentricity 
and  angle  of  advance. 

In  Fig.  15  draw  0</and  oa  at  right  angles  to  each  other. 


FIG.  15. 

With  o  as  a  centre  and  the  lap  as  a  radius,  draw  the  lap- 
circle  kqg.     Draw  ok  as  the  crank  position  for  the  point  of 


OVERTRAVEL   AND   PROBLEMS, 


21 


cut-off.  Lay  off  mp  equal  to  tb  ^  steam-lead.  The  problem 
then  becomes,  to  pass  a  circle  through  k,  <?,  and  /.  Bisect 
ok  and  op  by  perpendiculars  meeting  at  z.  z  is  then  the 
centre  of  the  valve-circle.  Draw  oz  and  the  valve-circle. 
aofis  the  angle  of  advance,  and  of  is  the  eccentricity. 

21.  Problem  3. — Given  the  cut-off,  angle  of  lead,  width 
of  port,  and  the  overtravel,  to  determine  eccentricity,  lap, 
lead,  and  angular  advance. 

In  Fig.  1 6  draw  oa  and  od  at  right  angles.     Lay  off  ok 


FIG.  16. 

for  the  point  of  cut-off  and  og  for  the  angle  of  lead.  As  the 
valve-circle  must  cut  the  lap  circle  on  these  lines,  the  centre 
of  the  valve-circle  must  lie  on  a  line  oe  which  bisects  the 
angle  kog.  After  drawing  this  line,  take  any  point,  as  z1 ',  for 
the  centre  of  the  trial  valve-circle.  Draw  the  circle  og'fk'. 
Draw  the  corresponding  lap-circle  k'q'g' .  Then  if  our  trial 
valve-circle  is  the  same  as  the  actual  one,  q'f  should  equal 
the  width  of  the  port  plus  the  overtravel. 

Suppose  q'f  is  less  than  the  width  of  the  port  plus  the 
overtravel.     Draw  any  line  through  /',  as  /'i,  equal  to  the 


22 


VALVE-GEARS. 


width  of  the  port  plus  theovertravel.  Join  \q' ,  and  through 
o  draw  02  parallel  to  qi  ;  then  f'2  is  the  actual  eccentricity. 
Lay  off  of  equal  to  f'2.  Bisect  of  at  z,  and  draw  the  valve- 
circle  kfpo.  Through  k  draw  the  circle  kqm.  Then  of  is 
the  eccentricity,  ok  is  the  lap,  mp  is  the  lead,  and/<?#  is  the 
angle  of  advance. 

22.  Problem  4. — Given  the  point  of  cut-off,  lead,  and 
port-opening,  to  determine  the  angular  advance,  lap,  and 
eccentricity. 

Draw  oa  and  od  at  right  angles  to  each  other,  Fig.  17, 


FIG.  17. 

and  ok  to  represent  the  point  of  cut-off.  Lay  off  ob  below  o 
on  the  line  ok  equal  to  the  lead,  and  bg  equal  to  the  port- 
opening.  Draw  gc  parallel  to  od,  and  make  ge  equal  to  the 
port-opening.  Join  oe.  Make  or  =  ob,  and  draw  qr  parallel 
to  od.  Draw  the  circle  oust,  with  oc  as  a  radius.  Lay  off 
st  =  cu,  and  draw  tof.  Then  aofis  the  angular  advance.  To 
find  the  diameter  of  the  valve-circle,  etc.,  proceed  as  in  the 
last  problem.  (See  solution  of  same  problem  as  given  in 
Fig.  100.) 

To  prove  our  construction: 

From  the  figure,  onf  and  ofp  are  right-angled  triangles. 
Calling  the  angle  aok  =  ft,  and  the  lead  equal  to  z,  we  have 


t 

OVERTRAVEL   AND   PROBLEMS.  23 

COS    *          ?    =  / 


and 

sin  d  =  l+i. 


Taking  the  value  of  /  from  the  first  equation  and  putting 
it  in  the  second,  and  reducing,  we  have 

\(i—p)  cos  ft}  cos  d  -\-  \p  —  (i  —  p)  sin  ft}  sin  d  =  i. 
Using  an  auxiliary  angle  0  such  that 

_  p-(i-p]  sin  ft  _  p  +  (p~  i)  sin  ft 

(i  _  p)  cos  ft  (i  -  p)  cos  ft 

we  have 


cos  <5       tan      sin  6  = 


—  i  cos  0 
cos  S  cos  0  +  sin  §  sin  0  =  .     _  ^  CQS~g» 

-  *  cos  0 


—  /  COS   0 

tf  —   0  -I-  COS  "'   f r- -. 

(p  —  t)  cos  ft 

Now  in  our  construction 
gc  —  (p  —  i)  sin  ft     and     ec  —  p  +  (p  —  i]  sin  ft, 

ec 
co  =  (p  —  i)  cos  ft,     and     tan   0  =  ZT^'     or 

0  =  1 80  —  eoc, 

—  i  cos  0 

cos  ~J  7—  —TT—       ^  =  —  <:^,     or 
(/  -  0  cos  ft 

d  =  1 80  —  eoc  —  cos  —  aof, 

as  by  construction. 

23.  When  the  Piston  and  Eccentric-Rods  do  not  travel 
on  Parallel  Lines. — The  simplest  method  of  connecting  up 
a  valve  has  heretofore  been  used.  Suppose  now  that  the 


24  VALVE-GEARS. 

connection  is  as  shown  in  Fig.  18.  In  this  figure  the  valve- 
face  is  not  parallel  to  the  direction  of  the  stroke  of  the  pis- 
ton, and,  secondly,  the  valve  is  driven  through  a  lever  which 
is  pivoted  at  a  fixed  point  a.  If  the  arms  ac  and  ab  are  un- 
equal, the  eccentricity  is  no  longer  equal  to  the  distance 
travelled  by  the  valve  on  either  side  of  its  middle  position. 
The  eccentricity  or  actual  throw  of  the  eccentric  should  not 

ab 
be  that  deduced  from  the  diagram,  but  should  be  —  X  r,  as 


FIG.  1 8 

this  value  would  cause  the  valve  to  move  a  distance  equal 
to  r. 

The  chord  of  the  arc  through  which  c  travels  should  be 
parallel  to  the  valve  face.  The  chord  of  the  arc  through 
which  b  travels  should  be  so  arranged  that  a  line  parallel  to 
the  chord  and  above  it  a  distance  equal  to  ^  the  versed 
sine  of  £  the  arc  should  pass  through  the  centre  of  the  shaft. 

24.  To  determine  the  Position  of  Eccentric. — To  deter- 
mine the  position  of  the  eccentric,  suppose  oe  to  be  the  line 
of  travel  of  the  piston.  Draw  the  line  db  as  the  line  of 
travel  of  the  end  of  the  eccentric-rod  ///;.  From  the  figure 
the  valve  will  be  in  mid  position  when  the  eccentric  is  at  og, 
or  directly  opposite.  If  the  crank  is  on  the  dead-point  of, 
and  it  is  desired  that  the  engine  turn  in  the  direction  of  the 


OVERTRAVEL   AND   PROBLEMS.  2$ 

arrow,  the  eccentric  should  be  so  set  that  its  motion  would 
continue  to  open  the  port  on  the  right  as  the  crank  leaves 
the  dead-point. 

That  is,  the  valve  must  move  towards  the  left,  c  must  move 
to  the  left,  b  must  move  to  the  right,  and  therefore  the  eccen- 
tric must  be  on  the  line  og  above  the  shaft,  as  it  is  only  in 
this  position  that  the  motion  of  the  crank  in  the  direction 
of  the  arrow  would  tend  to  open  the  port. 

Lay  off  the  angle  gom  equal  to  the  angular  advance  in 
the  direction  in  which  the  engine  is  to  turn.  Then  the 
angle  mo/is  the  angle  between  the  crank  and  the  eccentric 
for  the  engine  to  turn  in  the  direction  of  the  arrow. 

Therefore,  to  find  the  proper  position  of  the  eccentric 
with  respect  to  the  crank,  put  the  engine  on  one  dead-centre 
and  set  the  eccentric  so  that  the  valve  is  in  its  middle  posi- 
tion. If  direct  connected  set  it  ahead  of  the  crank,  and  if 
through  a  reverse  lever  set  it  behind  the  crank.  Then  move 
the  eccentric  through  an  angle  equal  to  the  angular  advance 
in  the  direction  in  which  the  engine  is  to  turn,  and  secure  it 
in  place. 

25.  Effect  of  changing  Dimensions. — An  examination  of 
Fig.  ii  will  show  how  a  modification  of  any  part  of  the 
valve  or  its  connections  will  affect  the  distribution  of  the 
steam. 

If  the  eccentricity  is  increased,  steam  is  admitted  earlier 
and  cut-off  later,  the  lead  and  the  overtravel  are  increased. 

An  increase  in  the  angular  advance  increases  the  lead, 
makes  admission  and  cut-off  earlier,  and  a  decrease  in  angu- 
lar advance  has  the  opposite  effect. 

Increasing  the  lap  lessens  the  lead,  makes  admission  later 
and  cut-off  earlier. 

QUESTIONS. 

25.  What  is  overtravel,  and  what  effect  has  it  on  the  dis- 
tribution of  steam  ? 

26.  How  determine  the  lap  and  lead  when  r,  d,  and  the 
point  of  cut-off  are  given  ? 


26  VALVE-GEARS. 

27.  How  determine  r  and  tf  when  the  lap,  lead,  and  the 
point  of  cut-off  are  given  ? 

28.  How  determine   the  lap,  lead,  and  angular  advance 
when  the  cut-off,  angle  of  lead,  arid  port-opening  are  given  ? 

29.  How  determine  the  angular  advance,  lap,  and  eccen- 
tricity when  the  point  of  cut-off,  lead,  and  port-opening  are 
given? 

30.  How  determine  the  angle  between  the  crank  and  ec- 
centric when  the  piston  and  valve  do  not  travel  in  parallel 
lines? 

31.  What  effect  on  distribution  of  steam  has  an  increase 
in  angular  advance  ?     A  decrease? 

32.  What  effect  on  the  distribution  of  steam  has  a  short- 
ened valve-stem  ? 

33.  What  effect  has  a  lengthened  eccentric-rod  ? 

34.  What  must  be  done  in  a  given  engine  to  increase  the 
lead? 

35.  How  in  a  given  engine  could  you  increase  the  lead 
on  one  end  and  at  the  same  time  decrease  it  in  the  other? 

PROBLEMS. 

8.  Given  lap  \  inch,  width  of  port  if  inches,  eccentricity 
2f   inches,  lead  ^  inch,  through    what  angle   is  the  crank 
moving  while  the  port  is  opening?     Had  the   eccentricity 
been  2^  inches,  through  what  angle  would  the  crank  move  ? 

9.  Given  steam-lead  \  inch,  cut-off  .8  stroke,  port-opening 
i \  inches,  to  determine  the  angular  advance,  lap,  and  eccen- 
tricity. 

10.  Given  lap  -J  inch,  lead  £  inch,  and  cut-off  at  .8  stroke, 
to  determine  the  eccentricity  and  angular  advance. 

11.  How  much   must  the  angular  advance  be  increased 
to  make  the  lead  f  inch  ? 

12.  How   much    must   the  eccentricity  be  increased  to 
make  the  lead  f  inch  ? 

13.  How  much  must  the  lap  be  changed  to  make  the  lead 
finch? 


OVERTRAVEL   AND   PROBLEMS.  2/ 

14.  Given  the  eccentricity  2§  inches,  8  =  30°,  and  cut-off 
.70  stroke,  required  the  lap  and  lead. 

15.  Given   the  cut-off  at  .8  stroke,  angle  of  lead  8°,  and 
the  port-opening  i£  inches,  required  the  lap,  lead,  and  angu- 
lar advance. 

16.  Given  r  =  2f  inches,  lap  =  J  inch,  stroke   24  inches, 
connecting-rod    60   inches.      Cut-off   takes   place    in   stroke 
towards  the  shalt  at  20  inches  :  at  what  point  does  it  take 
place  in  the  return  stroke?     How  far  has  the  piston  yet  to 
travel  in  each  stroke  when  the  port  opens  to  steam? 

17.  Given  steam-ports   if  inches,  outside  lap  f  inch,  ec- 
centric-rod  56  inches,  3  =  25°,  r  —  2\" ,  reverse-shaft  arms 
roj  to  eccentric-rod  and  nf  to  valve-stem,  which  is  above 
the  piston-rod.     Centre  of  reverse-shaft  7"  above  centre  line 
of  engine  ;  piston  travel  and  valve-face  are  parallel.     What 
is  the  angle  between  the  eccentric  and  crank  for  running  in 
each  direction? 


CHAPTER  IV. 

MODIFICATIONS  OF  THE  PLAIN  SLIDE-VALVE. 

26.  Double-ported  Valves. — It  sometimes  happens  that 
the  steam-port  necessary  to  give  sufficient  opening  must  be 
very  wide.  The  eccentricity  must  be  correspondingly  large 
to  cause  the  valve  to  open  the  port  fully.  In  such  cases  the 
valve  is  often  made  double-ported,  as  shown  in  Fig.  19.  In 


£17 

a    ^^M 

^ 

FIG.  19. 

this  figure  each  steam-passage  a  and  b  has  two  steam-ports 
opening  into  it.  The  steam  is  not  only  around  the  outside 
of  the  valve,  but  also  fills  the  passages  d  and  e,  which  ex- 
tend entirelv  through  the  valve. 

As  the  valve  moves  to  the  right  steam  passes  through 
both  ports  into  the  steam-passage  a,  while  on  the  left  steam 
passes  out  of  both  ports  into  the  space  under  the  valve,  and 
thence  to  the  exhaust.  In  designing  a  valve  of  this  kind 
the  problem  is  exactly  the  same  as  for  a  single  valve,  the 
only  care  that  must  be  taken  for  the  double-ported  valve 
being  to  see  that  the  opening  e  never  passes  over  the  left- 
hand  port  to  the  passage  a,  and  that  the  opening /never 
passes  over  the  right-hand  opening  into  the  same  passage. 

27.  Allen  or  Trick  Valve. — We  have  so  far  been  dealing 
with  valves  made  in  the  same  shape  as  that  shown  in  Figs,  i 
and  10,  but  this  form  is  modified  in  numerous  ways  to  fit 
the  varying  circumstances  under  which  it  is  to  be  used. 

28 


MODIFICATIONS   OF   THE   PLAIN   SLIDE-VALVE. 


29 


The  modification  used  on  many  locomotives,  and  known  as 
the  Allen  or  Trick  valve,  is  shown  in  Fig.  20.  a  and  b  are 
the  steam-passages,  and  c  the  exhaust-passage  in  the  cylin- 
der casting.  The  valve  is  a  single  casting  dd,  having  a  pas- 


sage e  cast  in  it. 


As  the  valve  moves  to  the  right,  when  the  point /of  the 
valve  passes  over  the  edge  g  steam  passes  under  the  right- 
hand  edge  of  the  valve  through  the  passage  e  into  the  steam- 


FlG.    20. 

passage  a,  while  at  the  same  time  steam  passes  by  the  end  i 
of  the  valve  which  has  moved  past  the  edge  h  of  the  port 
a,  allowing  steam  to  enter  through  double  the  area  for  the 
same  movement  that  could  be  obtained  with  the  ordinary 
valve.  Exhaust  takes  place  under  the  valve  through  the 
spaced,  as  in  the  plain  slide. 

When  high-pressure  steam  is  used  the  unbalanced  pres- 
sure on  the  back  of  the  valve  is  considerable,  causing  exces- 
sive wear,  thus  tending  to  make  the  valve  leak,  and  requiring 
considerable  power  to  move  it.  To  reduce  this  as  much  as 
possible  some  method  of  balancing  is  resorted  to,  and  in  the 
valve  just  described  strips  /,  /  on  the  back  of  the  valve  bear 
against  the  cover  of  the  chest  and  prevent  the  steam  from 


3°  VALVE-GEARS. 

passing  into  the  space  ;«,  thus  relieving  the  valve  of  the 
pressure  over  that  area. 

28.  Piston-Valves. — Piston-valves  are  used  for  the  same 
purpose,  a  modern  example  being  shown  in  Fig.  21,  which  is 
a  sketch  of  the  valve  used  on  the  Armington  &  Sims  En- 
gine. 

a  and  b  are  the  passages  leading  into  the  cylinder,  con- 
tinuations of  which  extend  entirely  around  the  valve.  g,g  are 
bushings  which  form  the  valve-seat,  which  are  cylindrical, 


FIG.  21. 

and  have  the  proper  openings  into  the  passages  leading  to 
the  cylinder.     The  valve  itself  consists  of  two  flat  flangec 
plates  c  and  k,  held  at  their  proper  distance  apart  by  the 
hollow  cylinder^/,  which  also  carries  two  discs,  e  and  /,  form 
ing  passages  next  the  plates  c  and  k.     Steam  fills  the  space 
h  inside  the  valve,  and  the  exhaust  takes  place  at  each  end 
of  the  valve  at  /,  i. 

As  the  valve  moves  towards  the  left,  steam  passes,  as 
shown  by  the  arrow,  from  the  space  h,  directly  into  the  left- 
hand  port,  and  also  into  the  passage  between /and  k  on  the 
right,  through  the  hollow  stem  d  into  the  passage  between 
c  and  e,  and  thence  into  the  port  also,  thus  making  the  valve 
a  double-ported  one.  The  exhaust  passes  through  the  pas- 
sage b  into  the  space  i  at  the  end  of  the  valve,  and  into  the 
exhaust-pipe. 

29.  Taking  Steam  Inside. — In  this  valve,  as  in  many 
others,  steam  is  taken  inside  instead  of  outside  ;  but  the  steam 
lap  herej  as  in  the  case  of  the  plain  slide,  is  the  distance  the 


MODIFICAl^IONS   OF   THE   PLAIN  SLIDE-VALVE. 


valve  must  move  from  its  middle  position  to  admit  steam. 
In  this  case  also  the  valve  must  move  in  the  opposite  direc- 
tion to  that  in  which  it  would  if  it  were  a  plain  slide.  The 
small  figure  would  therefore  show  the  method  of  connecting 
the  eccentric  and  crank  for  motion  in  the  direction  of  the 
arrow,  the  eccentric  being  directly  connected  to  the  valve. 

30.  Two  or  More  Valves. — To  separate  the  steam  and 
the  exhaust  passages  so  that  the  steam  entering  the  cylinder 
shall  not  come  in  contact  with  a  valve  partly  cooled  by  the 
exhaust,  and  oftentimes  to  get  a  better  distribution  of  steam 


/ 


'^^/^^y^^^/^  £?%• 

•i^J  '^^^V^y^       /fr-^j 

ft" 


;wy/y/y///y////y^^^ 


FIG.  22. 

and  exhaust,  two  valves  or  more  are  used.  These  valves  are 
usually  balanced  by  making  them  flat  plates,  and  allowing 
them  to  move  in  a  space  just  large  enough  to  allow  of  free 
movement  without  steam  leakage,  or  by  fitting  pressure* 
plates  on  the  back  of  the  valve,  the  position  of  which  can 
be  adjusted. 

Fig.  22  shows  the  four  valves  of  a  Porter  Allen  Engine, 
the  exhaust-valves  of  which  are  connected  to  the  same  stem, 
while  the  steam-valves  are  usually  connected  to  different 


32  VALVE-GEARS. 

sterns  driven  from  the  same  eccentric,  but  in  such  a  way  that 
each  valve  is  moving  its  fastest  when  opening  and  closing 
its  port. 

In  Fig.  22  steam  fills  the  space  a  and  extends  around  the 
pressure-plates  bb.  The  steam-valves  are  shown  at  c  and  d, 
and  consist  of  flat  plates  having  rectangular  openings 
through  them.  At  e  and /are  the  exhaust-valves,  which  are 
kept  on  their  seats  by  the  pressure-plates  gg.  Steam  passing 
from  a  into  the  left-hand  end  of  the  cylinder,  and  from  the 
right-hand  end  through  the  exhaust-valve  e,  is  shown  by  the 
arrows. 

If  these  valves  were  connected  directly  to  eccentrics  in 
the  small  figure,  the  steam-valve  would  be  connected  to  a 
as  it  moves  to  the  right  to  admit  steam  on  the  left  and  is 
therefore  directly  connected.  The  exhaust-valve  should  be 
connected  at  b,  as  this  valve  moves  to  the  left  to  exhaust  on 
its  right  and  it  must  be,  therefore,  indirectly  connected. 
Both  might  be  connected  at  a,  and  a  reverse  lever  used  to 
drive  the  exhaust-valves,  as  shown  in  Fig.  18.  The  actual 
method  of  driving  these  valves  is  shown  in  Fig.  53. 

QUESTIONS. 

36.  Make  a  sketch  of  a  section  of  a  double-ported  slide* 
valve.     Why  are  double  ports  used? 

37.  Sketch  an  Allen  valve.     What  are  its  supposed  ad 
vantages  ? 

38.  Why  are  piston-valves  used  ?     Sketch  a  piston-valve 
which  would  act  as  a  plain  slide. 

39.  How  does  the  valve  ot  the  Armington  &  Sims  engine 
work? 

40.  How  draw  the  Zeuner  diagram  for  a  double-ported 
valve? 

41.  What  change  is  necessary  in  the  setting  ol  an  eccen- 
tric if  steam  is  taken  inside? 

42.  Give  a  definition  of  steam-lap  which  would   cover 
all  valves  which  are  a  modification  of  the  plain  slide. 

4.3.  Why  are  separate  valves  used  for  steam  and  exhaust  ? 


t 
MODIFICATIONS   OF   THE   PLAIN  SLIDE-VALVE.  33 

44.  In  what  shape  are  flat  valves  made,  and  how  are 
they  kept  steam-tight? 

45.  Describe  the  valves  of  the  Porter- Allen  engine. 

PROBLEMS. 

1 8.  Given  the   maximum   port-opening  to  be  4  inches, 
double  ports  to  be  used,  cut-off  at  .85  stroke,  lap  i-J  inches, 
bridges  I \  inches,  exhaust-port  7  inches,  exhaust  closes  at  .9 
return  stroke.     Make  a  sketch  of  a  section  of  the  valve,  giv- 
ing all  the  dimensions  on  the  face  and  seat  of  the  valve. 

19.  In  a  single-ported  piston-valve  taking  steam  inside, 
r  =  2f  inches,  d  =  30°,  lap  =  £  inch,  exhaust-lap  £  ",  steam- 
port  i-J  inches,  bridge  i  inch,  exhaust-port  3  inches,  make  a 
sketch  showing  the  valve  in  the  position  it  would  occupy 
when  GO  =  30°. 

20.  One  of  the  exhaust-valves  of  a  four-valve  engine  is 
3J  inches  outside  and  2^  inches  inside.     The  exhaust-port  is 
I J  inches  and  the  exhaust-lap  J".     If  this  valve  is  driven  by 
an  eccentric  having  r  =  i£  inches,  and  set  with  30°  angular 
advance,  what  is  the  crank  position  for  opening  and  closing 
the  port  to  exhaust  ?     What  is  the  angle  between  the  crank 
and   the  eccentric?     How  prevent  steam  escaping  as  the 
valve  moves  the  other  way? 

21.  One  steam-valve  of  a  four-valve  engine  moves  over  a 
port    if  inches  wide.     The  walls  of   the  valve  are  \  inch 
thick,  and  the  cut-off  is  to  take  place  at  f  stroke,  with  \"  lap. 
If  r  —  if  inches,  make  a  sketch  of  the  valve  when  GO  —  120°, 
and  indicate  the  direction  of  its  motion. 


CHAPTER  V. 


EQUALIZING  CUT-OFF,    LEAD,   COMPRESSION,    AND 
RELEASE. 

31.  Equalizing  Cut-off. — Our  diagram  gives  only  the 
crank  position  corresponding  to  the  points  of  cut-off,  admis- 
sion, etc.  If  the  corresponding  position  of  the  piston  is  laid 
down  as  shown  in  Fig.  6,  it  will  be  found  that  steam  is 
admitted  for  a  longer  time  in  the  end  of  the  cylinder 
away  from  the  crank,  than  in  the  end  towards  it,  the 
difference  being  due  to  the  angularity  of  the  connect- 
ing-rod. One  way  by  which  this  can  be  overcome  is 
by  varying  the  lap  on  the  two  ends  of  the  valve. 
Thus  m  Fig.  23  the  piston,  connecting-rod,  and  crank  are 


FIG.  23. 

shown,  and  the  crank  positions  ie  and  if  for  cutting  off  in 
both  ends  of  the  cylinder  at  the  same  point  in  the  stroke  are 
laid  off 

That  is,  the  engine  turns  in  the  direction  of  the  arrow, 
and  the  distance  ac  —  bd.  Lay  off  the  valve-circles  as  shown 
at  igk  and  ilk.  The  point  g  in  which  the  line  le  cuts  the  upper 
valve-circle  gives  ig  for  the  lap  on  the  right  side  of  the  valve. 

34 


EQUALIZING  CUT-OFF,  LEAD.COMPRESSION,  AND  RELEASE  3$ 

The  point  //,  when  the  line  fi  cuts  the  lower  valve-circle, 
gives  us  hi  for  the  lap  on  the  left-hand  side  of  the  valve. 

It  must  be  remembered  that  while  we  have  made  the 
cut-off  equal  in  the  two  ends,  we  have  made  the  points  of 
admission  different  in  the  two  ends  of  the  cylinder  and  the 
leads  unequal.  In  exactly  the  same  way,  by  making  the 
exhaust-lap  unequal  on  the  two  ends  either  the  compression 
or  release  can  be  made  the  same  for  both  strokes. 

32.  Equalizing  Cut-off  and  Lead. — As  equal  cut-off 
means  unequal  lead  in  an  engine  as  ordinarily  connected,  the 
converse  is  equally  true,  that  if  a  simple  slide-valve  is  set  to 
give  equal  lead  on  both  ends  of  the  cylinder,  the  cut-off  will 
be  different  in  the  two  ends  of  the  cylinder,  and  any  change 
we  may  make  in  the  valve  or  in  the  eccentric  will  not 
remedy  it. 

It  is  possible,  however,  in  many  cases  to  make  both  the 
cut-offs  equal,  and  the  leads  practically  so,  by  modifying  the 
connection  between  the  eccentric  and  valve.  The  leads 
may  be  slightly  unequal,  but  the  angles  of  lead  or  the  dis- 
tance the  piston  has  to  travel  can  be  made  exactly  equal. 
Fig.  24  shows  how  this  may  be  done. 

Determine  from  the  valve-diagram  the  angular  advance, 
eccentricity,  lap,  etc.,  to  give  the  required  cut-off  on  one 
end.  Adding  90°  to  the  angular  advance  will  give  the  angle 
between  the  crank  and  eccentricity.  In  Fig.  24,  with  c  as  a 
centre,  and  a  radius  equal  to  the  half-travel  of  the  piston  or 
the  throw  of  the  crank,  draw  the  circle  drfek.  Make  ra 
equal  to  the  length  of  the  connecting-rod  and  ab  the  stroke. 
Let  g  be  the  point  of  cut-off  as  the  piston  travels  from  a  to 
b,  and  h  the  point  ol  cut-off  on  the  return  stroke.  With  g 
and  h  as  centres,  mark  the  position/  and  k  of  the  crank  at 
the  points  of  cut-off.  Let  cd  and  ce  represent  the  crank 
position  lor  the  opening  of  the  port.  Lay  off  from  cd,  cf,  ce, 
and  ck  an  angle  equal  to  that  between  the  crank  and  the 
eccentric  in  the  direction  in  which  the  engine  is  to  run,  giv- 
ing the  points  d',  f'y  ef,  and  k'  on  a  circle  whose  radius  is  the 
eccentricity. 


36  VALVE-GEARS.   ' 

Evidently  when  the  one  end  of  the  eccentric-rod  is  at  d1 
or  f  the  valve  must  be  at  the  same  point,  for  when  the 
eccentric  is  at  d'  the  port  is  opening  to  steam  on  the  right, 
and  at  /'  it  is  closing  to  steam  on  the  right.  Similarly,  if 
one  end  of  the  eccentric-rod  is  at  e\  the  other  end  must  be 
at  the  same  point  as  when  the  eccentric  end  of  the  rod  is  at  k'. 

Taking  the  length  of  the  eccentric-rod  as  a  radius,  draw 
arcs  with  d'  and  /',  and  e'  and  k'  as  centres,  and  call  the 
points  of  intersection  /  and  m.  Bisecting  the  line  ml  by  the 
perpendicular  np,  and  taking  any  point  p  in  this  line  and 


FIG.  24. 

drawing  a  line  pq  at  right  angles  to  cb,  gives  us  an  angle 
qpn.  If  we  make  a  bell-crank  lever  with  this  angle  between 
the  arms,  and  connect  the  arm  q  to  the  valve  and  n  to  the 
eccentric-rod,  the  arms  of  the  lever  being  of  such  a  length 
that  the  valve  moves  the  proper  distance,  the  cut-off  and 
lead  on  one  end  will  be  exactly  equal  to  that  on  the  other. 

To  determine  the  point/,  lay  off  nr  equal  to  nm  and  draw 
rt  equal  to  the  lap  and  at  right  angles  to  the  direction  of 
movement  of  the  valve-stem.  Through  ;/  and  /  draw  nq,  and 
at  q,  where  this  line  cuts  the  line  of  movement  of  the  valve- 
stem,  draw  pq  at  right  angles  to  that  line  or  parallel  to  rt. 
Then  /  will  be  the  desired  point,  as/^  :  np  : :  rt :  nr  or  as  Im  : 
2  times  the  lap.  Care  should  be  taken  that  the  eccentric-rod 
and  bell-crank  arm  do  not  come  too  nearly  in  one  straight 
line. 

33.  Equalizing  Exhaust  and  Compression. — In  exactly 
the  same  way,  by  marking  the  position  of  the  eccentric  for 


EQUALIZING  CUT-OFF,  LEAD,  COMPRESSION,  AND  RELEASE.  37 

exhaust  and  compression  on  both  ends,  points  similar  to  / 
and  m  in  Fig.  24  can  be  obtained,  and  the  exhaust  and  com- 
pression on  both  ends  made  alike.  If  these  points  should 
fall  in  such  a  position  that  the  same  arc  would  pass  through 
them,  and  also  the  points  /  and  m,  it  would  be  possible  to 
design  a  motion  that  would  give  equal  cut-off,  lead,  exhaust, 
and  compression.  Ordinarily  this  is  not  possible,  and  the 
radius //should  be  so  taken  that  the  arc  passes  through  the 
points  /  and  m,  and  as  near  as  may  be  to  the  points  founu 
for  equalizing  compression  and  exhaust,  the  point  p  being 
above  or  below  the  line  /;«,  depending  on  the  position  of 
these  points. 

34.  Circular  Diagram  for  determining  Movement  of 
Piston. — A  rather  more  convenient  method  of  laying  down 
the  piston  position  for  some  cases  is  shown  in  Fig.  25. 
Suppose  a  to  be  the  centre  of  the  shaft,  ab  the  crank  on  one 
dead-point,  and  be  the  connecting-rod.  With  a  as  a  centre 
draw  a  circle  with  ac  as  a  radius,  and  with  b  as  a  centre 
uraw  a  second  circle  with  be  as  a  radius.  Suppose  that  the 


FIG.  25. 


crank  remains  fixed  at  ab,  and  the  connecting-rod  turns 
about  b.  When  the  end  of  the  connecting-rod  reaches  any 
point  as  g,  the  angle  between  the  crank  and  a  line  to  the  end 
of  the  connecting-rod  \s  gac. 

The  distance  gh  is  the  amount  the  end  of  the  connecting- 


38  VALVE-GEARS. 

rod  has  moved  towards  the  centre  of  the  shaft  when  the 
angle  between  the  crank  and  piston  line  is  gac.  Now  draw 

another  circle  with  a  radius  ad,  less  than  ac  by  the  stroke: 
then  when  the  angle  between  the  crank  and  piston  travel  is 
cag,  the  piston  is  hg  from  one  end  of  its  stroke  and  gi  from 
the  other.  To  find  the  position  of  the  piston  for  any  position 
of  the  crank,  draw  the  line  ah  for  the  position  of  the  crank: 
then  hg  is  the  distance  from  one  end  and  gi  from  the  other 
end  of  the  stroke. 

If  the  position  of  the  piston  is  given,  the  crank  position 
can  be  found  as  follows  :  Taking  cd  as  the  stroke,  take  the 
point  j  as  the  position  of  the  piston  for  which  the  crank 
position  is  required.  Draw  an  arc  with  a  as  a  centre  and 
aj  as  a  radius,  until  it  cuts  the  circle  having  the  point  b  as  a 
centre  in  the  point  k.  ak  is  the  crank  position  desired. 

QUESTIONS. 

46.  On  which  end  of  the  cylinder  must  the  lap  be  the 
greater  for  equal  cut-offs  ? 

47.  If   a   plain  slide-valve   is   directly   connected   to   an 
eccentric,  and  is  set  for  equal  lead,  on  which  end  is  the  cut- 
off the  shorter  ? 

48.  A  double-ported  piston-valve,  taking  steam  inside,  is 
driven  directly  from  the  eccentric.    On  which  stroke  will  the 
cut-off  be  greater  if  the  valve  is  set  with  equal  lead  ? 

49.  What  effect  has  a  rock-shaft  on  the  equality  of  cut-off 
if  a  valve  is  set  with  equal  lead  ? 

50.  How  equalize  the  cut-off  by  varying  the  lap  ? 

51.  Could  the  cut-off  be  equalized  by  moving  the  valve 
on  its  stem  ?     If  so,  by  how  much  ? 

52.  Can  the  cut-off  and  /ead  be  always  equalized?     If  so, 
how  ? 

53.  Explain    the   method   of   determining   an   equalizing 
rock-shaft. 

54.  Is  it  possible  to  equalize  the  lead,  cut-off,  and   the 
opening  and  closing  of  the  exhaust? 

55.  How  can  a  circular  diagram  be  drawn  to  represent 
the  movement  of  the  piston  ? 


EQUALIZING  CUT-OFF,  LEAD,  COMPRESSION,  AND  RELEASE.  39 

PROBLEMS. 

22.  Given  r  =  2|  inches,  ports  i J",  bridges  i£",  exhaust- 
port  3-J",  tf  =  30°,  stroke  18  inches,  connecting-rod  45  inches. 
What  must  be  the  lap  to  cut-off  at  14  inches  on  each  stroke  ? 
If  the  exhaust  is  to  close  at   16  inches,  what  must  be  the 
exhaust-lap  on  each  stroke  ?     Sketch  the  valve  in  its  central 
position. 

23.  Given   steam-port    if   inches,  r  =  2\   inches,  cut-off 
in  each  end  to  be  at  .8  stroke,  stroke  22  inches,  connecting- 
rod  86  inches,  and  eccentric-rod  65  inches,  the  distance  from 
the  centre  of  the  piston-rod  to  the  centre  of  the  valve-stem 
being  18  inches.     Required  the  position  of  the  centre,  angle 
between  the  arms  and  length  of  the  arms  of  an  equalizing 
lever,  to  make  the  cut-off  and  angles  of  lead  equal. 

24.  In  a  certain  engine  the  ports  are  if  inches  and  the 
steam-lap  -J  inch,  lead  f  inch,  exhaust-lap  o.     The  stroke  is 
24  inches  and  the  connecting-rod  90  inches.     The  valve  is 
set  for  equal  lead.     How  much  must  it  be  moved  on  its 
stem  to  equalize  cut-off?    Where  is  the  piston  when  the  port 
opens  and  closes  to  exhaust  and  steam  in  both  ends  of  the 
cylinder? 


CHAPTER  VI. 
DESIGNING   AND   SETTING  VALVES. 

35.  Designing  a  Plain  Slide-Valve. — In  designing-  the 
valve  and  connections  for  an  engine,  certain  data  are  fixed 
by  the  details  of  the  engine  itself.  Thus  the  distance  from 
the  centre  of  the  shaft  to  the  centre  of  the  ports  is  fixed. 
The  size  of  the  ports  depends  on  the  diameter  of  the  cylin- 
der and  the  number  of  revolutions  per  minute,  as  steam 
should  not  be  made  to  travel  faster  than  6000  to  8000  feet 
per  minute  through  the  steam-port,  or  than  4500  to  5000  feet 
per  minute  through  the  exhaust-port. 

A  good  rule  is  to  multiply  the  area  of  the  piston  in 
square  feet  by  the  stroke  in  feet,  and  by  the  number  ol 
strokes  per  minute,  and  divide  by  6000  for  the  area  of  the 
steam-port  in  square  feet. 

The  thickness  of  the  bridges  between  the  steam  and  ex- 
haust passages  is  determined  principally  by  the  thickness  ot 
the  cylinder  walls  and  other  adjacent  parts  of  the  casting, 
and  is  generally  about  the  same  thickness  as  the  cylinder. 
A  good  empirical  rule  is  to  make  the  bridges  .4" +  -5  the 
width  of  the  steam-port  in  inches. 

To  retain  always  an  opening  under  the  valve  equal  to 
the  area  of  the  steam-port,  the  width  of  the  exhaust-port 
should  be  at  least 

r  -\-  .5  steam-port  +  exhaust-lap  —  .4  inches. 

The  amount  of  lead  to  be  given  to  a  valve  is  entirely  a 
matter  of  experience,  and  must  be  assumed.  In  locomotives 
it  varies  from  o  to  J  inch,  being  about  f  inch  when  the  valve 
has  its  least  travel,  and  from  o  to  T3F  when  it  has  its  greatest 
travel.  In  marine  engines  the  lead  varies  from  o  to  \\ 

40 


I 

DESIGNING  AND   SETTING    VALVES.  41 

inches.     The  angle  of  lead  in  stationary  engines  is  from  o° 
to  8°,  and  in  marine  engines  should  be  not  over  10°. 

The  outside  lap  varies  from  £  to  I  £  inches  in  locomotives 
and  i  to  3^-  inches  in  marine  engines,  while  the  inside  lap 
varies  from  o  to  \  inch  in  locomotives,  and  o  to  i£  inches  in 
marine  engines. 

The  eccentricity  varies  in  locomotives  from  2  to  3  inches, 
while  in  marine  engines  5  to  6^  inches  is  about  the  average 

Assuming  data  for  a  particular  case  as  follows,  required 
all  the  dimensions  of  the  valve  and  gear : 

Stroke  of  piston  ;  length  of  connecting-rod  ;  point  of  cut- 
off on  each  stroke ;  width  of  steam-port ;  width  of  exhaust- 
port  ;  steam-lead ;  distance  from  centre  of  piston-rod  to 
centre  of  valve-stem  ;  distance  from  centre  of  shaft  to  centre 
of  exhaust-port ;  point  of  exhaust  closure  on  both  ends  ;  thick- 
ness of  bridges  on  valve-seat ;  ports  to  be  on  top  of  cylinder, 
and  to  be  connected  through  an  ordinary,  or  an  equalizing, 
reverse-lever. 

36.  To  determine  Approximate  Solution. — In  Fig.  26 
draw  ab  and  cd  at  right  angles  through  o.  With  o  as  a 
centre,  with  any  convenient  radius  draw  ekf 'to  represent  the 
crank-pin  travel,  and  lay  off  fg  to  represent  the  position  of 
the  piston  at  cut-off.  Draw  gh  at  right  angles  to  ab,  and  draw 
oh  to  represent  the  position  of  the  crank  at  cut-off.  Continue 
this  line  to  i  so  that  oi  equals  the  lead.  Make  if  equal  to  the 
maximum  opening  of  the  port.  Draw/£  parallel  to  ob  and 
equal  to  the  port-opening,  and  join  ok.  Make  04  =  the  lead 
:=  oi,  and  draw  4 — 5  parallel  to  ob  until  it  cuts  the  arc  drawn 
with  oS  as  a  radius.  Lay  off  the  arc  8 — 6  from  5  to  7,  and 
draw  <?/  and  produce  it  to  /.  Then  col  is  the  angular  ad- 
vance. 

Take  any  point  as  s  as  a  centre,  and  draw  the  trial  valve- 
circle  cutting  ol  in  m.  With  o  as  a  centre,  draw  the  trial 
lap-circle.  Measure/^.  As  it  should  be  equal  to  the  port- 
opening,  draw  any  line  mq  through  m  equal  to  the  port- 
opening.  Join/  and  q,  and  draw  or  through  o  parallel  to/^. 
mr  is  the  eccentricity.  With  half  mr  as  a  radius,  describe 


VALVE-GEARS. 


the  valve-circle  ynx.  Draw  the  lap-circle  yx.  Then  zob  is 
ttie  angle  of  lead,  ox  is  the  lap,  on  is  the  eccentricity,  and 
con  the  angle  of  advance  which  we  will  use.  (The  method 
shown  in  Fig.  100  can  be  used  to  find  these  quantities.) 


FIG.  26. 


Find  02  the  point  of  closing  of  the  exhaust,  and  03  is  the 
exhaust-lap.  A  section  can  now  be  drawn  through  the  valve 
and  ports  as  shown  in  Fig.  27.  From  the  data  of  the  engine 


FIG.  27. 

we  have  ab,  be  and  ac,  and  ef  and  dc.  Make  fg  and  dh  the 
steam-lap,  and  ej  and  ci  the  exhaust-lap.  The  rest  of  the 
valve  can  then  be  drawn  in,  taking  care  that  the  exhaust- 
steam  in -passing  under  the  valve  into  the  exhaust-port  has 
fully  as  much  area  to  pass  through  as  in  the  passages. 


t 

DESIGNING  AND   SETTING    VALVES  43 

The  length  of  the  valve  over  all  is  df  -f-  twice  the  steam- 
lap.  The  valve-chest  must  be  at  least  long  enough  to  allow 
the  valve  to  move  from  its  middle  position  a  distance  equal 
to  the  eccentricity  in  either  direction.  The  length  of  the 
chest  inside  must  be  greater  than  df -\-  twice  the  lap  -f-  twice 
the  eccentricity.  The  valve-stem  must  project  through  one 
side  of  the  chest  when  the  valve  is  at  its  farthest  distance 
from  that  side,  and  the  length  of  the  stem  can  be  determined. 
The  distance  from  the  centre  of  the  shaft  to  the  centre  of 
the  exhaust-port  less  the  length  of  the  valve-stem  can  be 
taken  as  the  length  of  the  eccentric-rod.  A  reverse-lever  of 
equal  length  on  each  arm,  and  supported  midway  between 
the  centre  of  the  piston  and  the  centre  of  the  valve-stem, 
would  give  the  proper  reversal. 

If  a  valve  is  connected  up  as  just  determined,  the  eccen- 
tric-rod taking  hold  of  the  valve-stem  through  the  reverse- 
lever,  the  conditions  of  the  problem  would  be  approximately- 
solved,  and  ordinarily  this  would  be  sufficient ;  but  if  exactly 
equal  cut-off  and  lead  on  each  end  is  required,  the  following 
is  the  method  of  proceeding  : 

37.  Equalizing  Lever. — The  more  nearly  the  figure  is 
drawn  full  size,  the  more  nearly  accurate  the  dimensions 
will  be,  as  many  of  the  circles  intersect  at  very  acute  angles, 
allowing  considerable  error  if  drawn  to  small  scale. 

Draw  the  circle  ecb,  Fig.  28,  with  radius  equal  to  the 
throw  of  the  crank,  and  a'c'd'  with  radius  equal  to  the  eccen- 
tricity. Make  bg  the  length  of  the  connecting-rod,  and 
gh  =  the  stroke.  Lay  off  the  angles  aob  and  eod  ==  the  angle 
of  lead  from  Fig.  26,  and  make  gj  and  hi  equal  to  the  pis- 
ton travel  to  the  point  of  cut-off.  From  j  and  z,  with  the 
length  of  the  connecting-rod  as  a  radius,  mark  the  points  c 
and  /,  and  draw  the  lines  oc  and  of.  Lay  off  points  on  the 
eccentric-circle  so  that  aoa' ,  jof,  coc' ,  and  dod'  are  equal  to 
90°  minus  the  angular  advance,  as  a  reverse-lever  is  to  be 
used,  and  are  laid  off  behind  the  crank.  With  a'  and  c'  as 
centres,  and  radius  equal  to  the  length  of  the  eccentric-rod, 
draw  two  arcs  intersecting  at  /.  Similarly,  from  /'  and  d'  as 


44 


VALVE-GEARS. 


centres,  draw  arcs  in- 
tersecting at  k.  In  a 
similar  way  lay  off  the 
angles  eox  and  wob 
equal  to  the  exhaust- 
angle  of  lead  as  shown 
in  Fig.  26.  Make  hs  — 
gt  •=.  the  piston  posi- 
tion for  exhaust- 
closure,  and  mark  the 
positions  u  and  v  on  the 
crank-circle. 

Lay  off  the  points 
u',  v' ,  w',  and  x'  so  that 
the  angles  uou ,  vov' , 
etc.,  are  equal  to  aoa! '. 
From  w'  and  u\  with 
the  eccentric  rod  as  a 
radius,  mark  the  point 
n,  and  from  x'  and  v' 
mark  m.  Bisect  kl  by 
the  line  yp,  and  select 
a  point  p  on  it  such 
that  pk  will  describe 
an  arc  which  is  as  close 
to  m  as  to  n. 

Draw  qq'  at  the 
given  distance  from  hb, 
and  parallel  to  it,  for 
the  position  ot  the 
valve-stem.  As  this 
distance  is  given  in  our 
data,/  should  be  so  se- 
lected that,  while  the 
end  ot  the  eccentric- 
rod  moves  from  k  to  /, 
the  end  of  the  valve- 


DESIGNING  AND   SETTING    VALVES.  45 

stem  moves  twice  the  lap  from  q  to  r.  The  arms  of  the 
reverse-lever  should  make  an  angle  of  ypz,  and  the  arms 
should  be  pk  and  pr  inches  long.  The  centre  of  the  bell- 
crank  should  be  op'  from  the  centre  of  the  shaft  and  pp' 
above  it.  Our  valve-stem  mnst  be  determined  exactly  for 
the  reverse-lever. 


SETTING   THE    VALVE. 

38.  To  put  the   Engine  in  the  Centre.  —  Having  the 
engine  near  the  centre,  turn  it  away  from  the  centre  about 
fifteen  degrees.     Put  a  centre-punch  mark  on  the  frame  in 
such  a  position  that  a  tram  will  readily  reach  to  a  turned 
portion  of  the   shaft  or  wheel.     Mark  with  the  tram  a  line 
on  the  wheel,  and  scribe  a  mark  across  the  cross-head  and 
guide.     Turn  the  engine  past  the  centre  we  are  working  for 
until  the  mark  on  the  cross-head  and  guide  again  exactly 
correspond.     Do  not  turn  it  past  and   bring  it  up  again, 
Again  mark  with  the  tram  on  the  wheel. 

Bisect  the  distance  between  the  tram-marks,  and  turn  the 
wheel  until  the  point  midway  between  the  marks  is  just  the 
length  of  the  tram  from  the  fixed  point  on  the  frame,  and  the 
engine  will  be  on  the  centre.  The  other  centre  can  be  fixed 
in  the  same  way. 

39.  To  Set  the  Valve.  —  It  is  first  necessary  to  adjust 
the  valve  on  its  stem  and  then  to  place  the  eccentric. 

With  the  engine  on  one  centre,  set  the  eccentric  about 
in  the  right  place,  a  little  ahead  if  anything,  to  have  the 
ports  well  opened.  Measure  the  lead.  Turn  the  engine  in 
the  direction  it  is  to  run  to  the  other  centre,  and  again 
measure  the  lead.  Move  the  valve  on  the  stem  half  the 
difference  of  the  leads  on  the  two  ends,  so  that  if  again  tried 
they  would  be  alike.  Now  move  the  eccentric  on  the  shaft 
far  enough  to  close  the  port  and  then  back  until  the  proper 
auiount  of  lead  is  showing.  Secure  the  eccentric  in  position, 
and  the  work  is  done. 

In  putting  the   engine  on  the   centre,   the  lost  motion 


46  VALVE-GEARS. 

should  be  taken  up  so  that  the  distance  from  the  crank-pin  to 
the  cross-head  pin  is  the  same  on  both  sides  of  the  centre. 

In  taking  up  the  lost  motion  of  the  eccentric  and  valve- 
rod  it  should  always  be  done  in  the  direction  in  which  the 
engine  is  to  turn. 

QUESTIONS. 

56.  What  data  must  be  determined  before  the  valve  for 
an  engine  can  be  designed  ? 

57.  How  is  the  width  of  the  port  determined? 

58.  How  wide  should  the  exhaust-port  be  ? 

59.  What  should  be  the  thickness  of  the  bridges? 

60.  What  are  the  usual  limits  of  lap,  lead,  eccentricity, 
and  angular  lead  ? 

61.  Explain  in  detail   the  usual  method  of  determining 
the  parts  of  the  valve  and  connections. 

62.  How  is  the  equalizing  lever  laid  down  ? 

PROBLEMS. 

25.  An  engine  18  inches  cylinder  diameter  by  24  inches 
stroke  makes  125  revolutions  per  minute.     The  steam- ports 
are   15  inches  long.     Determine  width  of  steam-port,  thick- 
ness of  bridges,  and  least  width  of  exhaust-port. 

26.  An  engine   with  the    ports  as  in   Problem   25   is  to 
cut  off  at  .8  stroke  in  each  direction.     The  lead  may  be  be- 
tween -J  and  f  inch.     The  connecting-rod  is  60  inches  long, 
and  the  valve  is  on  the  side  of  the  cylinder.     The  exhaust- 
lead  is  to  be  -J  inch  on  both  ends.     Determine  all  the  other 
parts  of  the  motion. 

27.  An  engine  having  ports  as  above  is  140  inches  from 
centre  of  shaft  to  centre  of  exhaust-port.     The  vaive-stem  is 
40  inches  long.     The  valve  is  above  the  cylinder,  and  the 
valve-face  is  14  inches  above  the  centre-line  of  the  cylinder. 
If  possible,  with  equal  laps  the  cut-off  is  to  take  place  at  .8 
stroke,  and  admission  at  7°  before  the  dead-point.     Assume 
any  other  data  that  may  be  required,  and  design  the  entire 
motion,  giving  all  the  dimensions. 


CHAPTER  VII. 
THE   STEPHENSON   LINK. 

40.  The  Link. — The  arrangements  we  have  been  deal- 
ing  with  heretofore  have  been  for  the  purpose  of  causing  the 
engine  to  run  in  one  direction  only.  If  an  engine  with  an 
ordinary  slide-valve  had  two  eccentrics  attached  to  the  shaft, 
one  for  running  in  one  direction  and  one  for  running  in  the 
opposite,  so  that  either  could  be  connected  with  the  valve, 
the  engine  could  be  made  to  run  in  either  direction.  The 
ordinary  device  by  which  one  or  the  other  eccentric  actuates 
the  valve  is  called  a  link-motion.  The  one  probably  most 
commonly  used  is  called  Stephenson's  link,  a  centre-line 
sketch  of  which  is  shown  in  Fig.  29.  ab  is  the  crank,  ac  and 

I 


FIG.  29. 

ad  are  the  eccentrics,  ce  and  df  are  the  eccentric-rods,  and 
ef  is  the  link,  gh  is  the  valve-stem,  the  end  g  of  which 
moves  in  a  slot  in  the  link. 

Evidently  if  the  link  ef  is  lowered  so  that  e  comes  to  g, 
the  eccentric  ac  moves  the  valve,  and  the  engine  turns  in 
the  direction  of  the  arrow.  If  the  link  is  raised  the  whole 
way,  the  eccentric  ad  moves  the  valve,  and  the  engine  turns 

47 


48  VAL  VE-GEARS. 

in  the  opposite  direction.  At  any  intermediate  position  the 
valve  partakes  of  the  motion  of  both  eccentrics. 

The  link  is  caused  to  move  by  a  hanger,  as  kf,  attached 
to  a  point  on  the  link,  and  having  the  other  end  pivoted  to 
one  arm  kl  of  a  bell-crank  lever  klm,  so  that  by  moving  ;;/ 
to  the  right  or  left  the  link  is  raised  or  lowered. 

41.  Point  of  Suspension. — As  the  object  of  this  arrange- 
ment is  simply  to  raise  and  lower  the  link,  and  as  Jf  moves 
in  an  arc  of  a  circle  around  k,  the  point  on  the  link  to  which 
the  hanger  is  attached  should  be  such  that  it  will  not  influ- 
ence to  any  great  extent  the  motion  derived  from  the 
eccentric.  This  result  can  only  be  obtained  by  attaching 
the  hanger  to  that  point  of  the  link  which  is  most  to  be  used. 

In  Fig.  30  suppose/'  to  be  the  point  in  the  link  at  which 


\ 


FIG.  30. 

the  link  is  mostly  to  be  used.  If  the  hanger  is  attached  at 
p',  when  the  link  is  lowered  to  move  the  valve  the  point/*' 
moves  in  the  arc  of  a  circle  with/'/£  as  a  radius,  the  point 
p'  moving  very  nearly  in  the  straight  line  ah,  and  all  the, 
motion  derived  from' the  eccentrics  being  transmitted  to  the 
valve.  Suppose  now  the  hanger  had  been  attached  at  ;/  on 
the  chord  of  the  link.  The  point  n  will  now  move  in  the 
line  ah,  and  the  point/',  to  which  the  valve-stem  is  attached, 
will  move  in  a  curve  like  that  shown. 

42.  Slip  of  Block. — It  is  evident  that  as  the  valve-stem 
is  constrained  to  move  in  the  line  ah,  the  link  must  slip  up 


THE    STEPHENSON  LINK. 


49 


and  down  as  the  shaft  turns, 
and  this  is  objectionable,  if 
excessive.  Ordinarily  the  link 
is  suspended  at  one  of  three 
points — the  lower  end  of  the 
link,  the  centre  of  the  arc  of 
the  link,  or  the  centre  of  the 
chord.  -  The  last  point,  al- 
though probably  the  most 
often  used,  is  not  a  good  one, 
as  there  is  always  slip  in  the 
link  in  every  position. 

A  series  of  experiments 
made  by  Prof.  Marks  shows 
the  amount  of  slip  for  varying 
positions  of  the  link  and  end 
of  the  hanger.  In  Fig.  31  four 
sets  of  diagrams  are  given  for 
a  link  of  the  kind  we  are  deal- 
ing with. 

In  set  A  the  hanger  had  its 
lower  end  attached  to  the 
centre  of  the  arc  of  the  link, 
and  the  slip  increases  both 
ways  from  the  centre  ;  the  set 
B  is  obtained  when  the  lower 
end  of  the  hanger  is  attached 
at  the  centre  of  the  chord  of 
the  link,  and  there  is  no  position 
of  the  link  at  which  the  slip  is 
zero  f  set  C  is  taken  with  the 
hanger  attached  to  the  bottom 
of  the  link,  the  slip  when  the 
block  is  on  the  upper  half  of 
the  link  being  excessive,  or 
great  enough  to  interfere  with 
the  proper  distribution  of 


VAL  VE-GEARS. 


steam ;  set  D  is  taken  with  the  link  suspended  on  the  arc 
half-way  between  the  centre  and  bottom  of  the  link,  and 
is  used  to  show  that  the  hanger  should  be  attached  at  the 
point  most  to  be  used  if  otherwise  practicable. 

43.  Radius  of  the  Link. — The  radius  of  the  link  is  usu- 
ally   taken   as   equal  to    the    length    of    the    eccentric-rod 
for  the  purpose  of  making  the  valve  move  equally  on  both 
sides  of  a  fixed  point,  no  matter  what  is  the  position  of  the 
link.     This  it  does  not  exactly  accomplish,  but  it  is  nearly 
correct.     The  radius  of  the  link  is  sometimes  made  equal  to 
ae,  Fig.  29,  when  the  link  is  in  its  middle  position  and  the 
eccentrics  are  as  shown,  and  it  is  sometimes  made  equal  to 
qe,  the  point  q  being  half-way  between  c  and  d,  as  shown. 

Practically  there  is  no  advantage  in  taking  any  one  of 
these  three  lengths  as  far  as  keeping  the  centre  of  the  move- 
ment of  the  valve  at  the  same  point. 

44.  Kinds  of  Links. — The  links  ordinarily  used  are  rep- 
resented in  Fig.  32,  in  which  A  is  the  form  usually  used  on 


•©) 


FIG.  32. 

locomotives.  The  eccentric-rods  are  connected  at  d  and  e. 
The  hanger  is  attached  to  a  saddle  /,  and  the  end  of  the 
valve-stem  is  connected  to  a  rectangular  block  g,  called  the 
link-block,  sliding  in  the  slot. 

Another  form  of  the  slotted  link  is  shown  at  B,  which  is 
connected  to  the  eccentric-rods  on  the  ends  instead  of  at  the 


THE   STEPHEN  SON  LINK.  51 

side,  as  in  A.    Larger  eccentrics  and  a  longer  link  are  required 
than  with  form  A  to  get  the  same  movement  of  the  valve. 

The  form  often  used  on  marine  engines  is  shown  at  C. 
Two  similar  bars  hh  are  held  apart  by  the  studs  d,  e,  to  which 
the  eccentric-rods  are  secured,  to  an  extension  of  one  of 
which  the  hanger  is  usually  fastened. 

QUESTIONS. 

63.  Why  is  a  link-motion  ever  used  ? 

64.  Make  a  sketch  of  a  Stephenson's  link. 

65.  Where  on  the  link  should  the  suspension-rod  of  the 
link  be  attached  ? 

66.  What  effect  has  attaching  the  link  to  a  point  on  the 
chord? 

67.  What  is  meant  by  slip? 

68.  How  is  slip  affected  by  attaching  the  suspension-rod 
at  different  points  on  the  link  ? 

69.  What  is  the  radius  of  the  link  ? 

70.  Why  is  the  link  curved  at  all? 

71.  Sketch  the  different  forms  of  link. 


CHAPTER  VIII. 
THE   VALVE-DIAGRAM. 

45.  Travel  of  the  Valve. — In  our  work,  as  ac  is  or  should 
be  small  compared  with  ce,  we  will  assume  that  e  always 
moves  in  a  straight  line  through  a  and  e,  and  also  that  the 
angle  eah  is  practically  constant  for  any  position  of  the  link, 
during  one  revolution  of  the  shaft.  We  will  assume  that 
the  chord  of  the  link  is  2<r,  and  that  the  distance  the  link  is 
raised  or  lowered  from  its  middle  position  is  —  or  +  u. 

In  Fig.  33  call  ac  —  r  =  ad,  ce  •=.  g  =  dr,  er  =  2c,  and 
no  =  rio'  =  u.  '  It  is  now  required  to  find  the  distance  the 
valve  has  moved  from  its  central  position  when  the  link  is 
lowered  a  distance  u,  and  the  crank  has  moved  through  an 
angle  GO  from  the  dead-point.  In  the  figure  the  light  lines 
show  the  position  of  the  entire  arrangement  for  u  =  o, 
co  =  o,  and  the  heavy  lines  when  the  link  is  lowered  u,  and 
the  crank  has  moved  GO,  e'  practically  moves  in  ae' ,  and  call 
hae'  =  y.  If  the  angle  of  advance  is  #,  the  distance  e'  has 
moved  from  its  central  position  is  given  by  dropping  a  per- 
pendicular c's  from  c'  on  ae' .  Then  as  is  the  distance  e'  has 
moved  from  its  central  position.  Now  as  =  r  cos  c'ae'  = 
r  cos  (90  —  GO  —  6  —  y)  from  the  figure. 

Assuming  that  for  the  present  the  lower  eccentric  has 

not  moved,  the  movement  of  n'  to  the  left  would  be  — ~, 

times  as  much  as  e'  or  -     -  r  cos  (90  —  GO  —  d  —  y).    But  the 

movement  of  e'  has  been  along  ae' ,  so  that  the  point  n'  has 
moved  horizontally  or  along  ah  only  cos  y  times  this  dis- 
tance, or 

— —  r  COS  (90  —  GO  —  d  —  y)  COS  y.  .     .     .     (A) 

52 


THE    VALVE-DIAGRAM. 


Similarly  we  could  have  considered  that  the  point  r*  had 
moved  a  distance  equal  to  at  (when  /  is  the  foot  of  a  perpen- 


54  VAL  VE-GEARS. 

dicular  from  d'  on  ar')  to  the  left  of  its  middle  position. 
Calling  har'  =  y' ,  we  have 

at  =  r  cos  (90  —  y'  —  d  -{-  a*), 
and  this  would  have  caused  ri  to  move  to  the  left  a  distance 


'  -  X  at  X  cos  y'   °r    -  r  cos  (go—  y'  —  d  -\-co)  cos  y'.  (B) 

If  then  we  had  supposed  that  first  both  eccentric-rod 
ends  e'  and  r'  were  in  their  middle  positions,  and  first  e'  and 
then  r'  had  moved  to  the  position  shown  by  the  heavy  lines 
in  Fig.  33,  the  valve  would  have  moved  a  distance  equal  to 
the  sum  of  (A)  and  (B),  or 

x  =  -      -  r  cos  (90  —  oo  —  6  —  y)  cos  y 

c  —  u 
H  --  r  cos  (90  —  y'  —  S  -{-  GO)  COS  Y'  i 

or 

x  =  r  sin  (GO  +  d  +  y}  cos  y 

.    (C) 


From  the  figure,  as  y  and  y'  are  small  angles, 


c  —  u     .       ,       c-\-u 
cos  y=  i,  cos  y  =  i,  sin  ;/  =    —  —  ,  sin  p  =  -  . 

<S  o 

Expanding  equation  (C)  and  putting  in  the  above  values, 
ive  have 

f  =  —  (2  --  cos  d  cos  GO-\-  2c  sin  tf  cos  &?  -|-  2#  cos  d  sin 

o 

/  .  <;2  —  u*  \       ur 

=  r  cos  GO  I  sin  ^  -|  ---  cos  d  j  -|  --  cos  o  sm  co  ; 


THE    VALVE-DIAGRAM.  55 

which  gives  the  distance  the  point  n  has  travelled  from  its 
middle  position.  This  movement  is  practically  transmitted 
to  the  valve,  and  we  can  call  it  the  Distance  the  valve  has 
moved  from  its  central  position. 

46.  The  Valve-diagram. — We  have  seen  that  for  a  single 
eccentric  the  movement  can  be  represented  by 

x  =  r  sin  (GO  -f-  #)  =  r  sin  GO  cos  d  -}-  r  cos  GO  sin  #  ; 
or  if  we  call  r  cos  tf  =  A  and  r  sin  d  =  B,  we  have 
x  =  A  sin  GO  -\-  B  cos  GO. 

In  this  equation  A  and  B  are  evidently  the  coordinates  of 

A          B 

the  point /in  Figs.  11,  14,  15,  or  16,  or  —  and  -  are  the  coor- 
dinates of  z,  the  centre  of  the  valve-circles.  Similarly  for  a 
Stephenson's  link,  if  we  call 

A  =  —  cos  d     and     B  •=.  r  (sin  d  -f-  -         -  cos 

we  have 

x  =  A  sin  GO  -\-  B  cos  GO  ; 

showing  that  the  valve  movement  can  be  represented  by  a 
circle  as  in  the  case  of  a  single  eccentric. 

If  in  any  Stephenson's  motion  we  have  r,  c,  tf,  and  g 
given,  for  each  value  of  u  we  can  determine  A  and  B  from 
the  equation,  and  can  draw  the  valve-circle  corresponding 
to  the  position  of  the  link,  and  then  determine  the  various 
points  of  opening  and  closing  of  the  ports. 

If  u  —  c,  then  Ac  =  r  cos  6  and  Bc  —  r  sin  d,  and  the  dia- 
gram is  exactly  the  same  as  for  a  valve  moved  by  a  single 

eccentric.     If  21  =  o,  A0  =  o  and  B0  =  r  f  sin  d  -f-  —  cos  #),  and 

for  values  of  u  between  c  and  o,  the  values  of  A  and  B  fall 
between  those  given. 


56  VALVE-GEARS. 

47.  Curve   of  Centres. — Calling  —  and  —  the  varying 

coordinates  of  the  centres  of  the  valve-circles  for  different 
values  of  u,  we  have  by  eliminating  u  from  the  values  of  A 
and  B  the  follovnng  equation, 


,   .             r2  cos2  6 
sin  6  -| cos  d  I  =  B, 

o 

which  must  be  the  equation  to  the  curve  in  which  the  centre 
of  the  valve-circle  moves  as  u  changes  in  value. 

As  this  equation  has  only  A*  in  it,  the  curve  whose  coor- 

A  D 

dinates  are  A  and  B,  or  —  and  — ,  is  a  parabola.     The  axis  of 

2  2 

B  is  the  principal  diameter  of  the  parabola,  and  we  have 
already  seen  that  it  passes  through  the  points 


48.  To  lay  down  the  Valve-diagram. — We  are  now 
in  a  position  to  lay  down  the  valve-diagram  for  any  value 
of  u.  In  Fig.  34  draw  ab  and  cd  at  right  angles  to  each 
other.  Lay  off  aof  =  tf,  make  of=  r,  and  draw  the  valve- 
circle  with  <?/as  a  diameter.  This  then  is  the  valve-diagram 
for  u  =  c.  From  z,  the  centre  of  this  circle,  draw  zg  at  right 
angles  to  cd  and  equal  to  g.  Draw  gh  parallel  to  cd  and  equal 

to  Cj  and  join  z  and  //.   Then  oi  —  -  sin  6  from  the  figure,  and 

if  :  gh\\  zi  :  zg, 
or 

,          .       cX  —  cos  d 

. .      gh  X  zi  2  cr 

ij  —  -  — ~  —  cos  tf, 


THE    VALVE-DIAGRAM. 


57 


and 


of  =  -  (sin  d  +  -  cos  rf    =  —  c  , 

\  ' 


2   \  ^  /  2 

for  &  =  o. 

A  parabola  can  now  be  drawn  through  z  and  /,  having  j 
as  a  vertex  andjc  as  the  principal  axis,  by  any  known  method  ; 


FIG.  34. 

or  an  arc  of  a  circle  having  its  centre  on  cd  and  passing 
through  z  and  j  is  close  enough  for  practical  purposes. 
Drawing  the  arc  as  shown,  we  can  find  the  centre  for  any 
other  value  of  u  as  follows : 

For  u  =  c,  ~  =  —  cos  d,  and  for  u  =  ul ,  -—  =  —  cos  tf, 

A          c 
and  consequently  —f-  —  — .     Then  to  draw  the  valve-dia- 

•**•  11  U  + 

gram  for  u  =  -,  make  ik  =  J  X  zi  and  draw  kl  parallel  to  cd. 
j 


58  VALVE-GEARS. 

Then  /  is  the  centre  and  ol  is  the  radius  of  the  valve-diagram 
for  u  =  — .     As  the  angle  gzh  only  is  required,  the  lines  zg 

and  gh  can  be  drawn  to  any  scale. 

49.  The  Virtual  Eccentric. — By  the  virtual  eccentric  is 
meant  the  eccentric  which  with  certain  given  values  of  8  and 
r  would  give  the  same  distribution  of  steam  as  the  entire 
link-motion,  and  the  valve-diagram  drawn  in  the  last  article 

with  ol  as  a  radius  is  the  virtual  diagram  for  u  =  — ;  or  if 

the   link-motion  was  replaced  by  a  single  eccentric  having 
r  =  2ol  and  tf  =  aol,  the  distribution  of  steam  would  be  the 

same  as  with  the  link  at  the  point  u  =  -. 

50.  Designing  the   Gear. — The   designing    problem   is 
somewhat  different.     It  may  generally  be  stated  as  follows : 
Given    ports,   maximum    point  of  cut-off,  lead  or  angle  of 
lead,  distance  from  the  centre  of  the  shaft  to  the  centre  of 
the  ports,  to  lay  down  the  motion.     The  diagram  for  n  =  c 
can  be  laid  down  from  the  data  given  as  already  shown  for 
a  valve  with  one  eccentric,  and  the  lap  and  value  of  r  de- 
termined. 

51.  Valve-stem    and    Eccentric-rod. —  The    maximum 
length  of  valve-stem  should  then  be  determined  from  draw- 


FIG.  35. 

ings  of  the  steam-chest  and  ports.  This  taken  from  the  dis- 
tance between  the  centre  of  the  shaft  and  the  centre  of  the 
exhaust-port  is  the  distance  from  the  centre  of  the  shaft  to  the 
middle  position  of  the  centre  of  the  arc  of  the  link.  Thus,  in 
Fig.  35,  a  and  b  represent  the  two  positions  of  the  link  when 
the  crank  is  on  the  dead-points,  and  the  distance  we  have 


THE    VALVE-DIAGRAM.  59 

just  determined  should  be  the  distance  oc  to  a  point  half-way 
between  a  and  b.  This  distance  is  practically  g,  and  in  de- 
signing may  be  taken  so,  any  slight  error  being  corrected  in 
setting  the  valve,  as  the  valve-stem  should  always  be  ad- 
justable. 

52.  Length  of  Link. — To  determine   c,  the  half -link   is 
rather  a  question  of  experience  than  of  calculation.     The 
chord  of  the  link  should  be  longer  than  the  distance  between 
the  centres  of  the  eccentrics,  or  it  would  be  in  line  with  the 
eccentric-rod   at  some    time    during  the  revolution  of  the 
crank.     Perhaps  a  fair  value  would  be  c  =  \ r  to  3^.     We 
have  now  all  the  data  necessary  to  draw  the  valve-diagram 
for  any  value  of  u. 

53.  The  Hanger. — There  is  one  other  point  to  be  con- 
sidered, and   that  is  the  position  of  the  upper  end  of  the 
hanger  by  means  of  which  the  link  is  moved,  and  this,  of 
course,  depends  on  the  point  of  attachment  of  the  link  to  the 
hanger.    Suppose,  first,  that  it  is  attached  at  the  centre  of  the 
arc  of  the  link.     In  Fig.  35  we  have  seen  that  the  centre  of 
the  link,  when   the  engine  is  on  the  dead-points,  is  equally 
distant  from  c ;  that  is,  for  the  central  position  of  the  link  the 
centre  of  the  hanger  should  be  directly  over  c,  which  is  at 
a  distance  g  from  0,  and  at  a  distance  h  =  the  length  of  the 
hanger  above  ob.     It  would  be  better,  however,  to  lay  off 
such  a  distance  above  c  that  the  arc  described  by  the  lower 
end  of  the  hanger  would   be  equally  above  and  below  the 

I       l         Q\ 

line  abt  that  is,  the   distance   above  c,  h,  =  h  1 1  —  —  vers  —  ] 

\  2  2] 

when  h  =  length  of  the  hanger,  and  0  =  arc  described  by  the 
hanger. 

According  to  Zeuner,  the  upper  end  of  the  hanger  should 
move  in  the  arc  of  a  parabola,  which  can  for  all  practical 
purposes  be  replaced  by  the  arc  of  a  circle.  In  this  particu- 
lar case  we  have  taken,  the  arc  should  have  its  centre  directly 
over  the  centre  of  the  shaft,  and  at  a  distance  h  above  it,  and 
the  radius  should  be  g,  the  length  of  the  eccentric-rod. 

Actually,  this  is  impracticable;  and  a  convenient  way  of 
laying  it  down  is  as  follows : 


6o 


VALVE-GEARS. 


In   Fig.  36  let  ab  be  the  line  in  which  the  valve-stem 
moves,  a  being  the  centre  of  the  shaft.     From  a  lay  off  ac 

* 


FIG.  36. 

equal  to  the  length  of  the  hanger.  With  c  as  a  centre  and 
g  as  a  radius,  draw  the  arc  dfe.  Make  ge  =  gd  =  c.  Then 
efd  is  the  arc  in  which  the  end  of  the  hanger  should  move. 
This  would  require  that  one  arm  of  the  bell-crank  lever 
should  be  equal  to  g.  Assume  that  the  greatest  lever  allow- 
able is  equal  to  the  distance  hi.  From  any  point  h  in  cf,  with 
///  as  a  radius,  draw  klj.  Make  kj  =  de.  Mark  the  centre  of 
//at  ;//,  and  of  gf  at  n.  Move  h  to  the  right  a  distance  mn 
to  h' ;  then  h'  is  the  centre  for  the  bell-crank.  The  arc  with 
h'  as  a  centre  and  hi  as  a  radius  will  then  correspond  with 
the  arc  dfe  at  two  points,  and  will  be  nearly  right  through 
the  rest  of  its  motion. 

If  it  is  known  that  the  engine  is  to  run  mostly  with  the 
link  in  one  position,  the  corresponding  point  in  dfe  should 
be  determined,  and  the  point  h'  so  fixed  that  the  arc  with  hi 
as  a  radius  would  pass  through  that  point  on  dfe. 

54.  Link  Suspended  at  Bottom  or  Centre  of  Chord.— 
If  the  link  is  suspended  at  the  bottom,  exactly  the  same  con- 
struction will  hold  good,  except  that  instead  of  using  an  arc 
lying  equally  on  both  sides  of  cf,  it  should  lie  all  below  this 
line,  or  gd  should  be  2c.  If  the  link  is  suspended  at  the  cen- 
tre of  the  chord,  the  point  c  should  be  moved  in  the  figure 
to  the  left  of  the  line  ac  a  distance  equal  to  the  distance  from 
the  point  of  suspension  on  the  link  to  the  arc  of  the  link,  or 
the  distance  ad  in  Fig.  35. 


THE    VALVE-DIAGRAM.  6 1 

55.  Open  and  Crossed  Rods. — As  shown  in  Fig.  33,  the 
gear  is  said  to  be  with  open  rods.  Had  the  eccentric-rod 
from  c  been  attached  to  r,  and  that  from  d  to  et  the  gear 
would  have  been  with  crossed  rods.  In  other  words,  with  the 
crank  on  the  dead-point  away  from  the  link,  the  gear  is  said 
to  be  with  open  or  crossed  rods,  as  the  eccentric-rods  stand 
apart  or  are  crossed  when  viewed  as  in  Fig.  33. 

All  our  reasoning  and  formulae  apply  to  crossed  rods  as 
well  as  to  open  rods  by  putting  for  c,  —  c.  The  travel  of 
the  vaJve  becomes 

/  .  c*  —  u*  \       ur 

x  =  r  cos  a?  1  sin  o  —  -         -  cos  o  J  —  —  cos  d  sin  GO. 
\  eg  I        c 

In  Fig.  34,  c  being  negative,  gh  should  be  laid  off  to  the  left 
of  2g,  and  2 If  curves  the  opposite  way.  The  link  must  now 
be  raised  to  turn  the  engine  in  the  direction  of  the  arrow  in 
Fig.  33,  and  lowered  to  turn  in  the  opposite  direction. 

From  an  inspection  of  Fig.  34  it  is  evident  that  the  lead 
increases  as  the  value  of  u  becomes  less,  that  is,  with  open 
rods  the  lead  increases  as  the  link  is  moved  from  full  to  mid 
gear  and,  with  crossed  rods,  the  lead  decreases  from  ful)  to 
mid  gear. 

QUESTIONS. 

72.  What  is  the  equation  for  the  movement  of  the  valve? 

73.  What  approximations  are  made  in  determining  the 
movement  of  the  valve? 

74.  What  is  the  valve-diagram  for  the  Stephenson  link 
when  u  has  a  given  value  ? 

75.  What  is  meant  by  the  curve  of  centres?     What  is 
this  in  the  Stephenson  link,  and  how  can  it  be  replaced  by  a 
circle  on  the  diagram? 

76.  Explain  fully  the  method  of  laying  down  the  Zeuner 
diagram  tor  the  Stephenson  link. 

77.  What  is  meant  by  the  virtual  eccentric?     Sketch  a 
Stephenson  link-motion  and  the  virtual  eccentric  by  which 
it  might  be  replaced. 


62  VALVE-GEARS. 

78.  How  determine  the  length  of  valve-stem  and  eccen- 
tric-rod for  a  given  engine  ? 

79.  What  should  be  the  length  of  the  link? 

80.  How  lay  down  the  curve  in  which  the  upper  end  of 
the  hanger  or  suspension-rod  shoald  move? 

81.  When  the  link  is  suspended  at  the  top  or  bottom 
how  much  of  the  arc   moved  through  by  the  arm  of  the 
reversing-shaft  or  tumbling-shaft  should  be  used? 

82.  What  is  meant  by  crossed  rods  ? 

83.  How  draw  the  diagram  for  crossed  rods? 

84.  Make  a  sketch  with  the  crank  on  the  centre  towards 
the  link  of  an  engine  with  crossed  rods. 

PROBLEMS. 

28.  Given  diameter  of  cylinder  1 8  inches,  stroke  22  inches, 
connecting-rod  88  inches;   port-opening  3  inches,  cut-off  when 
u  =  c  at  .8  stroke  in  one  end  ;  length  of  link  13  inches,  eccen- 
tric-rod 66  inches,  6  =  20°.    Draw  the  Zeuner  diagram  and 
determine  the  point  of  cut-off,  when  u  =  3  and  4^  inches,  in 
both  ends  of  the  cylinder,  the  lap  to  be  the  same  on  both 
ends. 

29.  Eccentric-rod  50  inches,  connecting-rod  S2J-  inches, 
stroke  22  inches,  angular  advance  16°,  cut-off  for  //  =  c  at  .85 
stroke,  2c  =  \2\  inches,  r  =  3".     What  must  be  the  lap  on 
each  end  of  the  valve  and  what  the  lead  and  cut-off  when 
u  =  4  inches  with  crossed  rods. 

30.  In  the  valve-diagram  for  the  last  problem  lay  down 
the  parabola  for  the  curve  of  centres,  and  determine  in  inches 
the  actual  difference  in  cut-off  between  that  given  for  the 
circular  curve  of  centres  and  that  lor  the  parabola  tor  u  —  4 
inches,  on  both  strokes,  the  lap  to  be  as  determined  in  the 
last  problem. 

31.  Given  r  =  4$  inches,  eccentric-rod  76  inches,  2c  =  24 
inches  for  a  bar  link.     12  inches  is  the  greatest  movement 
of  the  link-block  in  the  link.     If  the  angle  of  advance  is  18°, 
what  is  the  crank  position  tor  cut-oft  ?     If  the  cut-off  is  to  be 
at  .7,  the  stroke  and  the  angle  of  lead  to  be  7°,  what  is  the 
angular  advance? 


CHAPTER  IX. 
EQUALIZING   LEAD   AND  CUT-OFF. 

56.  Equalizing  Lead. — The  change  in  lead,  referred  to 
in  the  last  article,  may  be  lessened  by  changing  the  angle 
of  advance,  or,  what  amounts  to  the  same  thing,  by 
making  the  angle  ot  advance  of  the  two  eccentrics  different. 
In  Fig.  37  (which  is  practically  Fig.  29  with  the  line  abr 
added),  if  ac  and  ad  are  the  eccentrics  in  the  same  relative 


FIG.  37. 

position,  the  crank  now  occupying  the  position  ab' ,  it  is  evi- 
dent that  the  crank  has  not  reached  the  dead-point  by  the 
angle  bob' . 

In  Fig.  38  we  have  the  valve-diagram  for  seven  positions 
of  the  link — three  for  running  in  one  direction,  three  for  the 
other,  and  one  for  the  link  in  its  middle  position,  or  in  mid- 
gear,  as  it  is  called.  Evidently  in  this  figure  the  line  ab  repre- 
sents the  position  of  the  crank  when  the  old  crank  ab,  Fig.  37, 
is  on  its  dead-point.  When  the  new  crank  ab'  of  Fig.  37  reaches 

63 


64 


VAL  VE-GEARS. 


the  dead-point,  the  old  crank  in  Fig.  38  has  moved  to  ab' ,  the 

angles  bab1  in  the  two  figures 
being  equal.  That  is,  ab'  cor- 
responds to  the  dead-point  of 
the  new  crank.  If,  therefore, 
we  turn  the  entire  diagram 
through  the  angle  bab' ,  we  have 
the  circles  in  the  positions  we 
are  accustomed  to  see  them,  ab 
is  again  the  dead-point,  and  the 
angle  of  advance  for  the  upper 
circle  has  been  increased  by 
bab' ,  while  the  angle  of  advance 
for  the  lower  circle  has  been 
decreased  by  the  same  amount. 
An  inspection  of  the  figure 
will  show  that  the  lead  does 
not  vary  so  much  in  the  upper 
valve-circles  when  ab'  is  the 

dead-point,  and  is  more  nearly  constant  whatever  the  posi- 
tion of  the  link  as  long  as  the  engine  turns  in  the  direction 
of  the  arrow  ;  but  it  will  also  be  seen  that  the  lead  is  much 
more  variable  when  running  in  the  opposite  direction.  If 
then  the  crank  is  turned  backwards  from  the  direction  in 
which  the  engine  is  intended  to  run  most  of  the  time,  the 
lead  going  one  way  is  more  nearly  equalized. 

This  angle  is  readily  obtained.     Let  it  be  cr;  then  from 
our  equation  for  the  movement   of  the  valve   for  GJ  —  cr, 


u  = 


/  .             c*  —  c?           \       c,r 
x  =  r  cos  cr  I  sin  tf  -) cos  d)  -\ cos  o  sm  cr. 

'or  GO  =  cr,  u  =  o,  we  have 
x  =  r  ( sin  d  -| cos  dj  COS  cr. 


I 

EQUALIZING  LEAD  AND   CUT-OFF.  65 

In  order  that  the  lead  may  be  constant,  these  values  of 
x  should  be  equal;  or  equating  and  reducing,  we  have 


TC  C  T 

—  -  cos  d  cos  <r  =  —  cos  d  sin  cr, 


or 


c. 

tan  a  =  -  . 
g 

That  is,  if  c1  is  the  maximum  distance  the  link  is  to  be  low- 
ered, the  crank  might  be  put  back  cr°  ;  or,  in  other  words,  the 
go-ahead  eccentric  could  be  set  with  an  angle  of  advance 
tf  -f-  cr  and  the  backing  eccentric  with  an  angle  of  advance 
d  —  cr.  Referring  to  Fig.  34,  it  will  be  seen  that  the  angle 
gzh  is  the  angle  cr. 

It  is  possible  to  make  the  lead  the  same  for  both  ends 
of  the  cylinder,  for  both  full  and  mid  gear,  by  altering  the 
radius  of  the  link.  Thus,  in  Fig.  39,  let  a,  b,  c,  and  d  be  the 
positions  of  the  eccentrics  corresponding  to  the  dead-points, 
and  eg  and  f  A  be  the  chords  of  the  link.  With  a  as  a  centre 
and  af  as  a  radius,  mark  the  position  k  ;  and  with  c  as  a  centre 
and  ce  as  a  radius,  mark  the  position  i.  Let  o  be  the  centre 
of  Im  and/  the  centre  of  ik,  then  op  is  the  rise  of  the  arc  of 
the  link  ;  or  making  Ir  and  mn  equal  to  op,  the  arc  of  the  link 
should  pass  through  e,  r,  and  g,  or  f,  #,  and  h.  This  would 
bring  the  valve  at  such  a  point  that,  whether  in  full  or  in  mid 
gear,  the  lead  would  be  the  same  for  the  two  ends  of  the 
cylinder,  but  it  would  still  vary  somewhat  between  these 
points. 

57.  Equalizing  Cut-off.  —  It  would  be  possible  by  using 
an  equalizing  lever  to  make  the  cut-off  exactly  the  same  on 
both  ends  of  the  cylinder  for  one  position  of  the  link,  but  the 
best  way  of  accomplishing  equalization  of  cut-off  is  by  find- 
ing tentatively  such  a  hanger  that  the  link  is  in  the  right 
position  when  cut-off  is  to  take  place.  If  we  can  equalize  it 
for  full  gear  and  say  for  cut-off  at  half-stroke,  or  when  u  —  c 


66 


VALVE-GEARS. 


and  u  =  — ,  it  will  be  practically  equal  for  the  points  between. 


lc  is  first  necessary  to  show  how  to  lay  down  the  link  ior 
any  position  of  the  crank. 


EQUALIZING  LEAD  AND   CUT-OFF. 


67 


58.  To  lay  down  the  Motion. — In  Fig.  40,  let  o  be  the 
centre  of  the  shaft,  oa  the  crank,  and  ob  and  oc  the  eccentrics 


FIG.  40, 

Let  d  be  the  desired  position  of  the  point  of  suspension  of 
the  hanger.  Then  with  c  and  b  as  centres,  and  the  length  of 
the  eccentric-rod  measured  to  the  point  of  attachment  to  the 
link  (which  may  or  may  not  be  g,  depending  on  the  kind  of 
link  used)  as  a  radius,  describe  arcs  g  and  h.  With  d  as  a 
centre  and  the  length  of  the  hanger  as  a  radius,  describe  an 
arc  ef. 

Cut  out  of  stiff  cardboard  or  soft  wood  veneer  a  template, 
shown  in  Fig.  41,  in  which  ij  is  the  arc  of  the  link, 
k  is  the  point  of  attachment  of  the  hanger,  and  /  and 
m  are  the  points  of  attachment  of  the  eccentric- 
rods.  Referring  to  Fig.  40,  if  this  template  is  put 
on  Fig.  40  so  that  /  falls  on  the  curve  g,  tn  on  the 
curve  h,  and  k  on  the  curve  ef,  the  arc  if  will 
then  show  the  position  of  the  centre  line  of  the 
link,  and  n  will  be  the  position  of  the  end  of  the 
valve-stem. 

59.  To  lay  down  the  Centre  of  the  Travel  of 
the  Valve. — Put  the  crank  on  each  dead-point  and 
the  line  Im  on  the  template  vertical,  and  find  positions  ^and  r 


A 


IJ 

FIG.  41. 


68  VALVE-GEARS. 

corresponding  to  n  of  Fig.  40.  A  point  s  half-way  between 
these  points  will  be  the  centre  of  travel  of  the  end  of  the 
valve-stem.  If  we  are  using  a  link  such  as  is  used  in  Fig. 
35,  the  point  c  of  that  figure  is  the  centre  desired.  The  dis- 
tance sn  is  the  distance  the  valve  has  moved  from  its  central 
position.  Now  sq  —  sr  is  the  distance  the  valve  has  moved 
from  its  central  position  when  the  crank  is  on  the  dead- 
points,  or  is  the  lap  plus  the  lead.  If  st  =  su  —  the  lap, 
the  arc  of  the  link  ij  must  pass  through  /  and  u  when  steam 
is  admitted  and  cut  off  from  the  cylinder,  because  the  valve 
will  then  have  moved  a  distance  equal  to  the  lap,  or  is  just 
ready  to  open  or  close  the  port. 

60.  To  determine  the  Centre   of   Suspension    of  the 
Hanger. — In  Fig.  40  put  the  crank  at  the  position  for  the 
desired    cut-off,  and  draw  curves  //  and  g  corresponding. 
Put  the  template  so  that  /  is  on  g,  m  on  h,  and  the  arc  ?/ pass- 
ing through  u.     Mark  the  position  of  k  as  in  Fig.  ^2.     Turn 
the  crank  to  the  corresponding  position  of  the  return  stroke, 
and  going  through  the  same   process,  but  making  ij  pass 
through  /  gives  another  point  k '.     Now  to  cut  off  equally 
on  both    strokes   the   lower  end  of   the    hanger  must  pass 
through  k  and  k' ,  or  its  centre  can  be  found  at  d  by  striking 
arcs  intersecting  at  d  with  k  and  k'  as  centres  and  the  length 
of  the  hanger  as  a  radius. 

If  now  we  find  several  points,  as  d ',  d^ ,  d^ ,  at  which  the 
upper  end  of  the  hanger  must  be  for  equal  cut-off,  say,  at  full 
gear,  and  when  cutting  off  at  half-stroke  running  in  both  direc- 
tions, we  have  four  points  through  which  the  upper  end  of 
the  hanger  must  move.  Finding  the  centre  of  the  circle 
through  d,  d' ,  d^ ,  d^ ,  gives  o  as  the  centre  of  the  reversing  or 
tumbling  shaft  and  <?/^as  the  length  of  the  tumbling-shaft 
arm. 

61.  Position  of  Stud. — The  point  of  attachment  of  the 
hanger  to  the  link  we  have  taken  as  some  distance  back  of 
the  arc  in  our  figure  simply  for  convenience  in  drawing.     In 
many  link-motions  as  used  on  locomotives  the  centre  of  the 
stud  is  so  located.    The  point  is  usually  determined  by  draw 


EQUALIZING  LEAD  AND   CUT-OFF.  69 

ing  a  centre-line  on  the  template  and  finding  the  position  of 
this  centre-line  for  cutting  off  at  each  half-stroke,  then  choos- 


FIG.  42. 

ing  points  k  and  k'  in  these  lines  so  that  kk'  is  horizontal, 
and  the  points  are  the  same  distance  from  the  arc  of  the 
link. 


VAL  VE-GEARS. 


62.  Reducing  Slip. — As  the  end  of  the  hanger  support- 
ing the  link  is  usually  constrained  to  move  in  the  arc  of  a 
circle,  and  the  tendency  of  this  point  when  not  over  the  link- 
block  is  to  move  in  a  figure  such  as  is  shown  in  Fig.  31,  the 
block  will  slip  in  the  link  a  greater  or  less  amount. 

A  number  of  expedients,  such  as  reducing  the  travel, 
altering  the  angular  advance,  or  lengthening  the  link,  are  re- 
sorted to,  thus  necessitating  a  complete  reconstruction  of  the 
gear.  A  suitable  choice  of  the  length  of  hanger  will  often 
reduce  the  slip  materially.  In  Fig.  43  we  have  the  centre- 
line of  the  link  shown  for  a  number  of  positions  of  the  crank. 
c  If  the  link  is  to  be  supported  at  the  bot- 
tom, the  curve  ab  is  the  path  in  which 
the  centre  of  the  link  moves  when  the 
link  is  the  whole  way  down  and  moves 
with  no  slip.  If  now  we  find  a  centre 
such  that  the  arc  de  is  the  closest  possi- 
ble to  the  curved  line  ab,  then  cd  is  the 
best  length  to  use  for  the  length  of  the 
hanger.  If  the  link  is  not  supported  at 
the  bottom,  a  figure  similar  to  ab  can  be 
drawn  from  the  actual  point  of  support, 
and  the  radius  found  from  that.  After 
the  length  of  the  hanger  and  centre  of 
tumbling-shaft  are  determined,  if  the  points  of  the  gear  inter- 
fere with  each  other  or  the  framing,  modifications  must  be 
made  in  the  determined  parts  to  remedy  the  trouble. 

63.  Error  of  the   Zeuner   Diagram. — In   deducing  the 
equation  for  the  Zeuner  diagram  we  have  made  certain  ap- 
proximations which  cause  the  diagrams  to  be  more  or  less 
inexact.     To  show  the  amount  and  position  of  these  errors, 
we  have  in  Fig.  44  drawn  the  valve-diagram  in  full  lines,  and 
the  actual  position  of  the   point  on  the  link  to  which  the 
valve-stem  is  connected  for  the  time  being  in  broken  lines, 
the  supposition  being  that  this  point  moves  exactly  in  the 
line  of  motion  of  the  valve-stem. 

It  will  be  seen  that  the  errors  are  least  while  the  piston 


FIG.  43. 


EQUALIZING  LEAD  AND    CUT-OFF. 


is  on  that  part  of  its  travel  away  from  the  shaft,  or  in  this 
case  when  the  cylinder  is  to  the  left  of  the  crank,  and  great- 
est while  on  that  part  of  its  travel  towards  the  shaft.  It  will 


Stroke  away  from  Shaft 


\ 

\ 


Stroke   toward  Shaft 


FIG.  44. 

be  seen  also  that  the  angles  of  advance  could,  in  the  case 
taken,  have  been  made  different  for  the  two  eccentrics  with 
advantage.  The  irregularity  of  the  diagram  also  makes  the 
point  of  cut-off  different  lor  the  two  ends  of  the  cylinder; 


72  VALVE-GEARS. 

that  is,  the  crank  positions  when  cut-off  takes  place  are  not 
180°  apart,  but  the  errors  of  the  diagram  are  such  that  the 
angularity  of  the  connecting-rod  when  two  and  one  half  to 
three  times  the  stroke  in  length  brings  the  piston  to  more 
nearly  the  same  distance  from  the  beginning  of  each  stroke 
when  the  valve  closes. 

QUESTIONS. 

85.  How  can  the  lead  be  made  more  nearly  constant  with 
a  Stephenson  link  ? 

86.  Through  what  angle  should  the  angular  advance  of 
the  eccentric  be  changed  to  make  the  lead  at  u  =  cl  and  u  =.  o 
the  same? 

87.  How  can  the  lead    be  equalized    by  changing   the 
radius  of  the  link? 

88.  How  determine  the  position  of  the  link  for  any  given 
value  of  u  and  GO  ? 

89.  How  can  the  centre  of  the  travel  of  the  end  of  the 
valve-stem  be  laid  down  ? 

90.  Explain  the  method  of  determining  the  arc  through 
which  the  upper  end  of  the  hanger  must  move  to  equalize 
the  cut-off. 

91.  Is  it  possible  10  make  the  cut-off  exactly  equal  v>r 
more  than  two  values  of  u  on  each  side  of  u  =  o? 

92.  Why  is  the  stud  often  placed  back  of  the  arc  of  the 
link? 

93.  How  can  the  point  be  determined? 

94.  Is  the  slip  greater  or  less  than  if  placed  on  the  arc? 

95.  Is  there  any  advantage  in  so  placing  the  stud? 

96.  How  determine   the   length   of   hanger  which   will 
reduce  the  slip  ? 

97.  In  the  Zeuner  diagram  what  are  the  errors,  and  is 
admission,  cut-off,  or  maximum  port-opening  most  affected 
thereby? 

98.  Do  the  errors  of  the  diagram  tend  to  neutralize  or 
to  increase  the  variation  in  cut-off  due  to  the  angularity  o* 
the  connecting-rod? 


EQUALIZING  LEAD  AND   CUT-OFF.  73 


PROBLEMS. 

32.  Given  in  a  Stephenson  link  with  open   rods  r  =  Si 
inches,  eccentric-rod  75^  inches,  2c  —  26  inches  ;  steam-lap  2±$ 
inches,  cut-off  for  u  —  c  at  .72  stroke.    What  is  tf,  and  to  what 
angles  should  d  be  changed  for  constant  lead  at  u  =  o  and 
u  =  c? 

33.  What  should  be  the  radius  of  the  link  \{2c=  12  inches, 
r  =  2\  inches,  length  of  eccentric-rod  =  56  inches,  and  the 
angle  of  advance  3  =  16°,  if  the  lead  at  full  and  mid  gear  is 
to  be  equalized. 

34.  In  a  Stephenson  link-motion  with  the  eccentric-rod 
attached  back  of  the  link  having  the  following  data  determine 
the  position  of  the  link  for  u  =  4  inches,  GO  =  40°  :  Eccentricity 
2j  inches,  angular  advance  18°,  eccentric-rod  56  inches  to  the 
arc  and  53^  inches  to  the  point  of  attachment  to  the  link: 
radius  of  the  link  55  inches,  20  =  \\\\"  ;  centre  of  tumbling- 
shaft  39"  from  centre  of  shaft  and  1  1  inches  above  centre  of 
the  engine  ;  length  of  hanger  14^  inches  ;  arm  of  tumbling- 
shaft  curved  so  that  when  the  link  is  in  its  middle  position 
the  upper  end  of  the  hanger  is  9^  inches  above  the  centre- 
line of  the  engine  arid  56  inches  from  the  shaft  ;  the  point 
of  suspension  of  the  link  being  T9¥  inch  back  of  the  centre  of 
the  arc,  stroke  24  inches,  and  connecting-rod  89^  inches. 

35.  With  the  above  data  find  the  centre  of  the  travel  of 
the  end  of  the  valve-stem. 

36.  With  the  above  data  as  to  the  HIIK,  and  with  the 
length  of  hanger  14^-  inches,  find  the  position  of  the   tum- 
bling-shaft and  length  of  the  arm  so  that  the  cut-off  is  equal 
for  both  ends  at  .5  and  .8  stroke. 

37.  Taking  the  link  data  as  above,  find  the  point  of  cut- 
off in  one  end  corresponding  to  -J  stroke  in  the  othei   end, 
and  after  changing  the  data  so  that  the  radius  of  the  link  is  56 
inches,  the  centre  of  suspension  is  at  the  centre  of  the  arc 
of  the  link,  and  the  upper  end  of  the  hanger  is  at  the  same 
point  as  before,  the  other  data  remaining  the  same,  examine 
the  cut-off  for  the  same  position  of  the  tumbling-shaft  arm 


74  VALVE-GEARS. 

and  determine  whether  the  changing  of  the  radius  of  the 
link  and  point  of  suspension  in  this  case  has  changed  the 
cut-off. 

38.  In   the   problem   thus  given  is   14^  inches  the  best 
length  of  the  hanger  that  could  have  been  taken  ?     If  not, 
determine  the  best  length  to  reduce   the  slip  as  much   as 
possible  for  full  gear,  the  link  to  be  supported  -f$  inch  back 
of  centre  of  arc. 

39.  Draw   a   Zeuner  diagram  for  Problem  34,    and  lay 

out  the  actual  movement  of  the  valve  for  u  —  c,  u  =  — ,  and 

u  =  o. 

40.  Replace  the  single  eccentric  in  Problem  27  by  a  link- 
motion  which,  when  in  full  gear,  will  fulfil  the  same  condi- 
tions.    The  length  of  the  hanger  is  not  to  be  greater  than 
2c,  and  cut-off  is  to  be  equal  at  .5  and  at  .8  stroke  for  the 
two  ends  of  the  cylinder. 


CHAPTER  X. 
THE  GOOCH  MOTION. 

64.  The  Gooch  Link.— Fig.  45  represents  a  centre-line 
diagram  of  a  Gooch  link-motion,  ab  is  the  crank ;  ac  and 
ad  are  the  eccentrics  set  at  equal  angles  of  advance ;  ce  and 
df  arc  the  eccentric-rods;  e/is  the  link  which  is  curved  in 


FIG.  45. 

the  opposite  direction  to  the  Stephenson  link  ;  kl\s  a  suspen- 
sion-rod which  supports  the  link  at  its  central  point  and  is 
attached  to  a  fixed  point  /  on  the  engine-frame ;  gj  is  the 
radius-rod,  the  end  g  sliding  in  the  slot  of  the  link,  and  the 
end/  being  attached  to  the  valve-stem.  The  end  ^is  moved 
up  and  down  in  the  link  by  means  of  the  hanger  km,  the 
lower  end  m  of  which  is  attached  to  the  radius-rod  and  the 
upper  end  h  is  attached  to  one  arm  hi  of  a  bell-crank 
pivoted  at  i. 

When  g  and  e  are  together,  the  eccentric  c  drives  the 
valve,  and  the  engine  turns  in  the  direction  of  the  arrow. 
When  the  radius-rod  is  lowered  so  that /and  g  are  together, 

75 


76  VAL  VE-GEARS. 

the  eccentric  ad  drives  the  valve,  and  the  engine  turns  in 
the  opposite  direction. 

65.  Movement  of  the  Valve. — The  method  followed  in 
deducing  the  equation  for  the  movement  of  the  valve  is 
identical  with  that  followed  for  the  Stephenson  link-motion. 
In  Fig.  46  the  crank  and  gear  are  represented  when  the 
crank  has  moved  an  angle  GO  from  its  dead-point  and  the 
radius-rod  has  been  raised  a  distance  u. 

The  distance  the  point  e  has  moved  from  its  central  posi- 
tion along  the  line  ae  is 

an  —  r  cos  (90  —  GO— d  —  y)  —  r  sin  (GO  +  d  -f-  y). 
The  distance  the  point  g  has  moved  corresponding  thereto  is 

r  (sin  (co-\-d  4-  y))  cos  y. 

In  the  same  way  the  point  f  has  moved  from  its  central 
position 

ap  =  r  cos  (90  —  d  —  y  -f-  GO)  =  r  sin  (d  -f-  y  —  GO) 
and  g  has  moved  a  corresponding  distance 


--  r  sin  (tf  +  y  —  a?)  cos  r, 


and  the  motion  of  g  from  its  central  position  is  therefore 
x  = r  sin  (GO  -\-  d  -(-  y)  cos  y  -| r  sin  (#  -f-  7—  G?)  cos  ;/. 

As  in  Fig.  45  the  radius- rod  gj  is  long  as  compared  to  gk 
or  ke,  the  horizontal  motion  of  j  is  practically  the  same  as 
of  g,  and  we  have  the  above  value  of  x  for  the  movement  of 
the  valve  from  its  central  position. 


THE   GOOCH  MOTION. 


77 


7$  VAL  VE-  GEARS. 

Expanding  the  second  member  of  this  equation,  and 

c 
ing  cos  y  =  i,  sin  y  =  -,  we  have 

x  =  r  f  -  cos  ^  -f-  sin  d\  cos  GO  -| I  cos  6 sin  tf)  sin  o>. 

This  equation  can  also  be  put  in  the  same  form  as  for  a  sim- 
ple valve.     If  we  let 

r  I  sin  tf  +  -  cos  tfj  =  A 

o 

and 

—  (costf--sind)=£, 

we  have 

x  =  A  cos  G?  -|-  ^  sin  00, 

which  is  the  same  equation  as  for  a  simple  valve. 

66.  Constant  Lead. — An  examination  of  the  equation  for 
x   will  show  that  the  value  of  A  is  constant  whatever  the 
value  of  u\  that  is,  for  every  value  of  u  the  valve-circle 
crosses  the   line  representing  the  dead-point  at   the   same 
point,  or  the  lead  is  constant. 

67.  Radius  of  Link. — When  the  engine  is  on  the  dead- 
point,  as   shown   in    Fig.  45,    if   the   lead   is   constant,  the 
point  j  must  not  change  position  as  the  radius-rod  is  raised 
or  lowered.     The  link  must  therefore  be  drawn  with/ as  a 
centre  and  jg  as  a  radius,  or  the   radius  of  the  link  is  the 
length  of  the  radius-rod.     The  eccentric-rod  and  the  radius- 
rod  should  each  be  as  long  as  possible,  but  a  better  distribu- 
tion of  steam  is  generally  obtained  by  making  the  eccentric- 
rod  the  longer.     The  same  statements  as  to  the  length  of  the 
link  apply  here  as  in  the  case  of  Stephenson's  link.     The 
distance  from  the  centre  of  the  shaft  to  the  centre  of  the 
exhaust-port  can  be  approximately  determined  as  follows : 
The  mean  position  of  the  chord  of  the  link  is  approximately 


THE   GOOCH  MOTION.  79 

at  a  distance  g  from  the  centre  of  the  shaft.     The  distance 
from  the  chord  to  the  arc  is  — ,  approximately.     The  dis- 

O  1 

tance  from  the  centre  of  the  shaft  to  the  centre  of  the  arc 
of  the  link  when  the  arc  is  in  its  middle  position  is*?-—  — -. 


Adding  to  this  the  length  of  the  radius-rod  g^  and  the  length 
of  the  valve-stem  to  the  middle  of  the  valve  rs,  we  have 


—     -  -\-gl 


=  the  distance  from  the  centre  of  the  shaft 


to  the  centre  of  the  exhaust-port.  This  is,  of  course,  only 
an  approximation,  but  it  is  correct  enough  for  practical 
purposes. 

68.  Suspension-rod.  —  The  point  of  support  of  the  sus- 
pension-rod for  the  link  should  be  in  such  a  position  that 
the  rod  swings  equally  on  each  side  of  a  vertical  line.  Fig. 


FIG.  47. 

47  gives  the  two  positions  of  the  link  when  the  crank  is  on 
the  dead-points. 

bd  —  be  -f-  cd  —  r  sin  d  +  g, 

approximately,  and 

be  =  —  ib  -\-te=  —  r  sin  d  -\- g, 

and  the  mean  position  is  at  £/=:£•  from  the  centre  of  the 
shaft.  As  the  hanger  is  attached  to  the  arc,  and  not  the 
chord,  the  distance 


80  VAL  VE-GEARS. 

If  then  the  suspension-rod  is  attached  to  the  link  at  the 
centre  of  its  arc  and  to  a  fixed  point  whose  distance  from  b 

f 
\s  g  — parallel  to  ad,  and  the  length  of  the  suspension- 

<5  i 

rod  above  ad,  the  attachment  will  give  the  proper  motion. 
The  distance  above  ad  is  usually  given  as  the  length  of  the 
suspension-rod,  but  this  brings  the  arc  in  which  the  centre 
of  the  link  swings  entirely  above  ad,  while  to  keep  the  mo- 
tion as  nearly  correct  as  possible  the  line  ad  should  inter- 
sect the  arc,  and  the  point  g  of  the  link  should  swing  equally 
above  and  below  this  line. 

69.  The  Hanger. — To  determine  the  arc  in  which  the 
point  h   of    Fig.  45    should  move.      From    the    point   //  in 
Fig.  47,  just  determined,  lay  off  the   length  of  the  radius- 
rod  to  the  right  in  the  figure.     Mark  on  this  distance  the 
point  m  of  Fig.  45.     Through  the  point  thus  determined 
lay  off  a  distance  mh  at  right  angles  to  ad.     Through  this 
point  draw  an  arc  whose  centre  is  on   the  right  side  of  the 
link  whose  radius  is  the  length  of  the  radius-rod,  and  whose 
centre  lies  at  a  distance  equal  to  the  length  of  the  hanger 
above  aj,  Fig.  45.     This  arc  should  lie  equally  above  and 
below  the  line  through  the  centre  of  the  arc,  parallel  to  aj\ 
and  is  the  curve  in  which  h  should  move.     Practicably  this 
is  much  too  large,  and  this  arc  would  be  replaced  by  one  of 
much  smaller  radius,  the  centre  for  which  would  be  deter- 
mined in  the  same  way  as  shown  for  a  Stephenson's  link  in 

Fig.  36- 

70.  The  Valve-diagram. — If    all  the  data  relating  to  a 
Gooch  motion  is  given,  the  valve-diagram  can  be  laid  down 
as  follows  :  Suppose  the  lap,  angle  of  advance,  throw  of  the 
eccentric,  length   of  link,  eccentric-rod,  radius-rod,  and  dis- 
tance to  the  centre  of  the  exhaust-port  be  given,  to  determine 
the  point  of  cut-off,  etc.,  for  a  given  value  of  u. 

Draw  ab  and  ac  in  Fig.  48  at  right  angles  to  each  other, 
and  lay  off  baf  =•  d.  Make  ad  =  c,  and  ae  =  g,  and  eh  =  r. 
Draw  the  perpendicular  ih  to  ae.  Make  the  angle  fak  =  dec, 
and  lay  off  ak  =  ei.  ak  is  then  the  diameter  of  the  virtual 


THE    GOOCH  MOTION. 


81 


eccentric  for  u  =  c.  Or,  make  af=  ryfak  =  dea,  and  draw 
fk  at  right  angles  to  af\  then  ak  is  the  diameter  of  the  virtual 
eccentric. 


FIG.  48. 
The  proof  of  the  construction  is  as  follows :  For  the  co- 


ordinates of  k  we  have,  from  page  78, 
A  =  aj  =  r  \^sin  d  -|-  —  cos  d J, 

o 

B  —  kj—r  (cos  <5  -•  -  sin  d], 


£• 


cos 


^  =  tan  akj  •=. 
r> 


- 

Q 


~         C        .  C 

cos  o sm  d       i tan 

S  g 


As  —  =  tan 


tan  #£    = 


tan  d  -f-  tan 


or 


i  —  tan  d  tan  dea1 
akj  =  6  -}-  dea, 
sin  akj  =  aj=r  fsi 


sn 


^ 

-  cos 
g 


82  VALVE-GEARS. 

=  r  sin  6  -\-  r  tan  dea  cos  tf 

r  sin  8  cos  dea  -f-  r  sin  dea  cos  tf 
cos 

sin  (dea  -\ 


sin  (dea r  +  #) 
cos  dea 

or 

sin  (dea  4-  d)  ? 

ak  =  r , rj 7-7  =  - 

cos  dea  sin  akj        cos 

#>£  =  r  sec  dfctf  ==  «', 

as  by  construction.     Draw  the  lap-circle  qvs. 

The  point  of  admission  for  u  =  c  is  then  on  aq,  and  of  cut- 
off on  as.  The  lead  is  rj.  For  any  other  value  of  u  lay  off 

JO"  Ji 

tne  point  g  so  that  -r  —  -.    Join  «  and  £-,  and  draw  the  valve- 

1  ft  ^ 

circle  with  agas  a  diameter;  au  will  now  be  the  point  of  ad- 
mission, av  the  cut-off,  and  the  lead  rj  will  be  the  same  as 
before. 

71.  To  Design  a  Gooch  Motion. — We  will  suppose  the 
same  data  to  be  given  as  in  a  Stephenson's  link  to  design  a 
Gooch  motion,  i.e.,  ports,  point  of  cut-off,  angle  of  lead,  dis- 
tance from  centre  of  shaft  to  centre  of  exhaust-port,  to  lay  down 
the  motion.  The  length  of  the  valve-stem  and  link  should  first 
be  determined  or  assumed.  From  the  distance  between  the 
centre  of  the  shaft  and  the  exhaust-port  should  be  taken  the 

length  of  the  valve-stem.     The  remainder  is  g-\-g,  -    - — . 

If  now  g  or  gt  is  assumed,  the  other  is  determined.  All  these 
data  must  be  fixed  before  trying  to  lay  down  the  diagram. 

In  Fig.  49  draw  ab  and  acat  right  angles,  and  let  as  and 
aq  be  the  positions  of  the  crank  for  cut-off  and  admission. 
Bisect  the  angle  saq  by  the  line  ak,  and  find  the  diameter  ak 
of  the  virtual  eccentric  for  the  required  port-opening,  as 
already  shown  in  Chapter  III.  Make  ad  =  c  and  ac  =  g,  and 


THE   GOOCH  MOTION.  83 

lay  off  ei  equal  to  ak.  Draw  ih  parallel  to  ab.  Then  eh  is 
the  eccentricity.  Make  kaf  =  dea.  Then  the  angle  fab  is 
the  angular  advance. 

We  have  now  all  the  data  required  for  laying  down  the 
motion,  except  the  hanger  for  the  link  and  radius-rod.     The 


FIG.  49. 

only  directions  that  can  be  given  about  them  are  to  make 
them  both  as  long  as  possible,  and  to  support  them,  as  pre- 
viously indicated. 

QUESTIONS. 

99.  Sketch  a  Gooch  motion  and  describe  it. 

100.  Deduce  the  equation  for  the  movement  of  the  valve, 
and  prove  that  the  Zeuner  diagram  represents  it. 

101.  How  does  the  lead  vary  with  a  Gooch  motion? 

102.  Why  is  the   radius  of  the   link  made  equal  to  the 
length  of  the  radius-rod  ? 

103.  How  determine    the  centre  of    suspension    for  the 
link? 

104.  What  is  the  curve  in  which  the  upper  end  of  the 
hanger  must  move  ? 

105.  Explain  the  method  of  drawing  the  valve-diagram. 

106.  What  data  must  be  given  in  designing  a  Gooch 
motion  ?     Explain  in  full. 


84  VAL  VE-GEARS. 


PROBLEMS. 

41.  Draw  the  Zeuner  diagram  for  the    Gooch   motion, 
having  8  =  20°,  r  =  2f  inches,  2c  —  12  inches,  £•  =  48  inches, 
lap  =  £  inch,  stroke  =  18  inches,  connecting-rod  45".    Deter- 
mine the  point  of  cut-off  for  u  =  o,  2,  4,  and  6  inches  on  both 
strokes. 

42.  If  in  the  above  problem  the  radius  of  the  link  is  36 
inches,  the  hanger  is  attached  10  inches  from  the  arc  and  is 
1 8   inches   long,  and  the  arm   of   the  tumbling-shaft  is  24 
inches,  the  centre  being  i/f  inches  below  the  centre-line  of 
the  engine  and  82  inches  from  the  shaft,  draw  the  motion 
when  oo  =  40°,  u  =  4  inches. 

43.  What  must  be  the  radius  of  the  link,  if  the  cut-off  is 
to  be  exactly  the  same  on  both  ends,  for  u  —  6  inches  ? 

44.  Replace  the  link-motion  of  Problem  40  with  a  Gooch 
motion  cutting  off  at  .8  stroke  on  both  ends,  using  such  other 
data  of  Problems  40  and  27  as  are  required,  but  making  the 
valve-stem  not  less  than  20  inches  long. 

45.  Taking  the  data  given  in  Problems  42  and  43,  show 
the  errors  of   the  Zeuner  diagram  for  u  =  4  and  u  —  —  6 
inches. 


CHAPTER   XI. 

THE   ALLEN   AND    FINK   MOTIONS. 

72.  The  Allen  Link-motion  is  represented  in  Fig.  50,  in 
which  ab  is  the  crank,  ac  and  ad  the  eccentrics,  de  and  £/the 
eccentric-rods.  The  link  fe  is  a  straight  one,  supported  at 


FIG.  50. 

its  centre  g  by  the  hanger  mg,  which  is  attached  to  one  end 
of  a  rocker-arm  mlk  turning  around  /.  hj  is  the  radius- 
rod  supported  by  the  hanger  kit  the  upper  end  of  which  is 
attached  at  k  to  the  other  end  of  the  rocker  mlk.  By  turn- 
ing mlk  about  its  centre  /,  the  link  is  lowered  at  the  same 
time  that  the  radius-rod  is  raised. 

73.  The    Valve- diagram. — The  diagram   for  the  valve- 
motion  is  the  circle,  as  in  the  case  of  the  other  links  that  we 


86  VAL  VE-GEARS. 

have  dealt  with.     The  movement  of  the  valve  can  be  deter- 
mined in  the  same  way,  and  is 


/   .  c*  —  uu.  \ 

=  r  (  sin  d  -(-  -  cos  6)  cos  GO 


.   ur  ( 

-{  --  ( 

c  \ 


c(u  —  u.} 
cos  6  —  -*—    —  -  sin  £    sin 


ug 

in  which  u  is  the  distance  the  link  has  moved  downwards 
from  aj,  while  ul  is  the  distance  the  end  of  the  radius-rod  has 
moved  upwards. 

The  hangers  ki  and  mg  should  be  of  equal  length,  and  as 
long  as  possible,  mk  should  be  equal  to  hi.  The  ratio  of 
the  parts  ml  and  Ik  is  given  by  the  equation 


and  as  Im  -\-  Ik  •=.  ai,  from  the  two  equations  the  value  of  Ik 
and  Im  can  be  found. 

The  position  of  /  can  be  found  by  laying  off  along  aj  a 
distance 


and  at  right  angles  to^/'a  distance  equal  to  mgthe  length  of 
the  hanger.  To  draw  the  valve-diagram  for  any  value  of  #, 
the  coordinates  of  the  centre  of  the  circle  are 

A       r  (  .  f-uu,  \ 

»==5(sin*  +  -— —  costfj 

and 

B      ur  I  c(u  — «,)    . 


ur  I 
-  =  — 

2         2C  \ 


gU 

For  any  value  of  u,  the  corresponding  value  of  ul  is 


T       I  ./    ^ 

"  ji  X  Im 


THE  ALLEN  AND   FINK  MOTIONS.  8/ 

The  motion  is  but  seldom  used  in  this   country,  and  it  is 
unnecessary  to  go  further  into  the  details. 

74.  The  Fink  Motion.  —  The  Fink  motion  is  represented 
in  Fig.  51,  in  which  R  is  the  crank,  od  the  eccentric  directly 
opposite,    the   travel  of   the  piston    being   supposed  to  be 
along  ob^.     dj\s  the  eccentric-rod,  rigidly  connected  at/  to 
the  link  cd  '.     The  point  p'  of  the  eccentric-rod  is  constrained 
to  move  in  the  line  ob0  ,  or  very  nearly  so,  by  the  suspension- 
rod  p'g  attached  to  a  fixed  point  on  the  engine-frame  at  g. 
mbf  is  the  radius-rod,  the  end  m  sliding  up  and  down  in  the 
link,  while  the  end  b'  is  attached  to  the  end  of   the  valve- 
stem  b'b0  .     The  radius-rod  is  moved  in  the  link  by  means  of 
the  suspension-rod  et,  the  lower  end  e  of  which  is  moved  in 
a  suitable  curve.     If  the  crank  is  on  a  dead-point,  the  point 
j  would  be  on  obQ  ,  and  nm  would  be  vertical. 

75.  Radius  of  Link.  —  As  it  is  desired  that  the  engine 
should    have  constant  lead  whatever   the    position    of   the 
radius-rod,  if  the  valve  does  not   change   us   position   the 
radius  of  the  link  must  be  mb'  ,  and  this  is  what  it  is  gener- 
ally made.     An  examination   of  the  figure  will  show  that, 
when  the  engine  is  on  one  dead-point,  the  distance  from  the 
centre  of  the  shaft  to  the  centre  of  the  valve  is 


and  when  the  engine  is  on  the  other  centre  the  distance  from 
o  to  b0  is 

_  od+dj  +  mb'  +  b'b,  =  -r  +  (a  +  b)  +&  +^2. 

That  the  lead  may  be  equal  for  both  ends  of  the  cylinder 
with  a  common  D  valve,  the  distance  from  the  centre  of  the 
shaft  to  the  centre  of  the  exhaust-port  should  be  the  mean 
of  the  two  values  of  ob0  above  found,  or 


VALVE-GEARS. 


THE  ALLEN  AND  FINK  MOTIONS.  89 

76.  Suspension  of  Link. — The  coordinates  of  the  point 
g  can  readily  be  determined.     On  one  dead-point 

ov  =  od  -\-  dp'  =  r  -f-  at 
and  on  the  other  dead-point 

ov'  =  —  od  -\-  dp  =  —  r  -f-  a, 

and  *&gpr  should  swing  equally  on  either  side  of  pg,  op  should 
be  the  mean  of  these  values  of  ov  or  a.  To  determine  the 
pointy,  lay  off  from  p  the  distances/^  •=.  pv'  =  r.  Draw  an 
arc  wj>'  with  the  desired  value  of '  pg  as  a  radius.  Bisect 
the  distance  pvz  at/\  (not  shown) ;  then  the  point  pl  should 
be  on  one  point  in  the  arc,  and  laying  off  p,g=  the  desired 
value  of  gp1  gives  g  the  centre  of  motion.  By  calculation, 


and 


Too  great  care  cannot  be  taken  in  laying  down  this  gear,  as 
the  approximation  to  the  movement  of  the  valve  is  much 
more  crude  in  this  case  than  in  any  that  have  preceded  it. 
77.  Movement  of  the  Valve. — From  the  figure 

ob.  =  of+fp'  +  p'k  +  kl  +  IV  +  b'b. , 
of=r  cos  oo,    fp'  =  a  cos  a,         p'k  =  (b  +  x)  cos  <*, 


kl  =  y  siri  a,      IV  =  Vg?  -  u\     b'b.  -  &  , 

and 

ob0  =  r  cos  co  -f  a  cos  a  -f-  (b  +  x)  cos  a  -\-  y  sin  a 

+St, '-'#+&• 


90  VALVE-GEARS. 

From  the  figure  also 

y  cos  a  =  u  -f-  (b  +  x)  sin  or, 


or 


»  -f-  (b  +  JT)  sin  «  .  u  sin  a  -|-  (£  -\-  x)  sin* 

and 


y  — — ,        y  sin  a  = 

J  cos  a  -* 


b  +  u  sin  a  -\-  x 

(b  -4-  x}  cos  a  -4-  y  sin  a  =  -  — . 

COS  a 

u* 

For  Vg?  --  i?  we  may  put  g^ ,  and  making  all  the 

substitutions  in  ob^ ,  we  have 

b  4-  u  sin  a 

obn  =  r  cos  oo  4-  a  cos  a  4-  - 


x  u^ 

'    cos  a:  "•  ^    '  ^2       2^-,  * 

As  ^'   is  an  arc  of  radius  g^ ,  approximately  y  =  2gjc, 
r  sin  GO  =  a  sin  or. 

We  have  seen  above  that 

_u-\-  (b  -{-  x)  sin  a 

COS  Of 

and  as  ^  sin  a  is  small,  we  can  make 

u  -4-  b  sin  a  y*        (u  -4-  b  sin  a}* 

V  —  —  and     x  =  —  =  — 

COS  Of  2g^  2gl  COS    a 

or 

b  +  #  sin  a 

obn  =  r  cos  GD  -\-  a  cos  #  + 

COS  a 

(u  +  6*nar   ,         ,  *1 

2^-,  COS3  «  2^" 

We  have  above     r  sin  G?  =  a  sin  #    and     sin  a  •=.  —  sin  &?, 


cos 


I         r*  r1    .  i  ra 

tf  =  A  /  i ^  sin2  &?  =  i  —  — ^  sina&?,  —    —  =  I  +  — sin2  ce?, 

y  a  20*          'cos«          '  2^2 


t 

ALLEN  AND  FINK  MOTIONS.  91 

and 

-  =  i  +  —, i  sin8  GO. 
'        2 


cos   a 
and 

rur      ubr\ 
oo  ^  =  r  cos  GO  -f-  f  —  -|- j  sin 


after  collecting  the  terms.  Now  the  movement  of  the  valve 
for  any  value  of  GO  is  the  distance  the  valve  has  moved  from 
its  central  position. 

We  have  already  seen  that  the  central  position  is  at 

ob,  =  a  +  £  +  £-,+£-,; 

the  movement  of  the  valve  is  therefore  the  difference  of  the 
last  two  equations,  or 


sin  GO 


.    ur(          b\ 
x  =  r  cos  co  -f-  —  ^i  -|-  —  j  si 

—~  a    sin3  GO.     (A) 


The  last  term  in  this  equation  is  generally  small,  and  may  be 
omitted  ;  and  then 


ur 
=  r  cos  GO  -4-  - 


ur  i          b\ 

—  ( l  +  — J  sin  a?, 


which  represents  the  movement  of  the  valve  for  any  value 
of  u  and  GO. 

78.  The  Valve-diagram. — The  coordinates  of  the  centre 

of  the  valve-circle  are  —  and  —  ( i  4- — );  so  that  if  we  have 

2  2a  \         gj 

the  data  of  the  gear  given,  the  point  of  cut-off,  admission, 
lead,  etc.,  can  be  determined  from  a  diagram  such  as  Fig.  52, 
which  represents  the  valve-diagrams  for  four  values  of  u. 
Evidently,  if  u  =  o,  the  diameter  of  the  valve-circle  is  r,  or 


VALVE-GEARS. 


the  lap  plus  the  lead  equals  the  throw  of  the  eccentric.  An 
examination  of  the  first  value  of  x  deduced  (A),  shows  that 
the  last  term  is  less  the  greater  the  value  of  a  and  the 


FIG.  52. 

smaller  the  value  of  b.  Generally,  b  is  made  very  small, 
oftentimes  zero,  or  the  point/'  in  Fig.  51  is  attached  directly 
to  j.  When  this  is  the  case, 


=  r  cos 


.    ur    . 
-|  ---  sm  GO. 


79.  Radius-rod  at  a   Fixed  Point   in  the   Link.  —  We 

have  supposed  in  our  deduction  that  u  is  kept  constant,  that 
is,  that  the  end  of  the  radius-rod  moves  in  the  link.  If  we 
had  supposed  that  y  was  constant  and  that  u  changed,  we 
would  have  found  that  the  same  equation  would  give  the 
movement  of  the  valve,  putting/  in  the  place  of  u. 

80.  Hanger   for   Radius-rod.  —  To  determine  the   point 
of  suspension  of  the  hanger  for  the  radius-rod.     Referring 
to  Fig.  51,  and  calling  tb'  =£•„,  we  have 


But  ts  =  -— ;  consequently 


I 
THE  ALLEN  AND  FINK  MOTIONS.  93 

We  have  already  seen  that  from  o  to  the  centre  of  the  port 
is  a  -\-  b  -\-  gi  +£"2  ;  consequently  the  position  of  s  when  the 
valve  is  at  the  middle  of  its  travel  is 


and  as  the  suspension-rod  should  be  vertical  for  this  point, 
for  the  centre  of  its  movement 


and 


he  —  et  —  st  =  h  —  —  u, 


when  h  =  length  of  suspension-rod.  If  we  call  y  =  oh  and 
y^  —  oh  =  a  -j-  b  -f-  gl  —  gQ  for  u  —  o,  and  z  =  he  and  z9  =  he 
=  h  for  u  —  <?,  we  have 


and 

z       z- 
*'  ' 

or  eliminating  u,  we  have 


This  equation  is  that  of  a  parabola  having  a  parameter 
of  2^0,  or  the  lower  end  e  of  the  suspension-rod  should  move 
in  a  parabola  as  the  radius-rod  is  raised  or  lowered.  As  this 
is  practically  impossible,  the  parabola  can  be  -replaced  by  a 
circular  arc  of  radius^,  the  vertex  of  the  arc  being  below 
the  line  obn  a  distance  h,  and  to  the  right  of  o  a  distance 


81.  Setting  the  Eccentric.  —  In  setting  the  eccentric,  as 
in  the  case  of  all  other  link-motions,  the  crank  should  be  put 
on  the  dead-point,  and  the  eccentric  should  be  put  so  that 


94  VALVE-GEARS. 

the  valve  is  in  the  middle  of  its  stroke.  The  eccentric 
should  then  be  moved  a  distance  equal  to  the  angular  ad- 
vance, in  this  case  90°,  and  secured  in  this  position.  The 
crank  and  eccentric  now  occupy  their  proper  relative  posi- 
tions. 

82.  Designing. — The  practical  designing  of  a  gear  of  this 
kind  is  largely  a  matter  of  experiment.     It  could  be  done 
in  this  way.     First  determine  or  assume  the  greatest  point 
of  cut-off  desired  and  the  amount  of  lead  or  angular  lead. 
Draw  the  valve-circle  for  u  •=.  its  greatest  value.     This  will 
determine  the  amount  of  lap.     Assume  that  b  =  o.     Take 
from  the  distance  between  the  centre  of  the  shaft  and  the 
centre  of  the  exhaust-port  the  length  of  the  valve-stem.    The 
remainder  is  a  -}- gl .     Now  ^  should  be  as  long  as  possible, 
and  ordinarily  a  must  be  of  considerable  length,  that  the 
link  may  clear  the  shaft. 

It  will  be  found  that  as  a  increases  the  value  of  c  increases 
also ;  but  in  our  demonstration  the  value  of  c  is  small  in  pro- 
portion to  gt ,  so  that  it  is  difficult  to  decide  as  to  the  propor- 
tions of  a  and  gt ,  and  the  only  directions  possible  are  to  make 
a  as  short  and  g1  as  long  as  possible.  We  will  suppose, 
however,  that  they  are  determined.  From  the  centre  of  the 
valve-circle  as  drawn  we  measure  the  distance  to  the  hori- 
zontal axis  of  the  figure.  This  distance  is  — .  The  value  of 

a 

r  is,  as  said  before,  the  lap  plus  the  lead,  so  that  we  have  the 

u  c 

value  of  -.     But  as  this  is  the  greatest  cut-off,  we  have  -; 
a  a 

and  having  the  value  of  a  determined  as  above,  the  value  of 
c  can  be  fixed.  The  lengths  of  the  suspension-rods  should 
be  as  great  as  possible,  and  are  generally  determined  from 
the  details  of  the  framing  of  the  engine,  or  other  details  hav- 
ing nothing  to  do  with  the  valve-motion  proper. 

83.  The  Porter-Allen  Motion. — The  only  application  of 
the  Fink  motion  commonly  found  in  the  United  States  is  the 
Porter-Allen  motion,  a  sketch  of  which  is  given  in  Fig.  53. 
In  this  figure  aw  is  the  centre-line  of  the  engine,     ab'  is  the 


THE   ALLEN  AND  FINK  MOTIONS. 


95 


-crank  in  one  position,  and  ac  is  the  corresponding-  position 

of  the  eccentric,     c  b  is  the  eccentric-rod  rigidly  attached  to 

the  link  bgf.     The  point  b  is 

attached   by  the  rod   be  to  a 

fixed    point  e  on  the   frame. 

h'i'  is  the   radius-rod  which 

moves  the  steam-valves. 

There  are  two  of  these 
steam-valves,  one  for  each 
end  of  the  cylinder,  as  shown 
in  Fig.  22.  The  valve-stems 
are  joined  by  the  rods  u'l' 
and  v'k'  to  the  arms  jl'  and 
jk'  of  a  rocking-shaft  /,  a 
third  arm//'  being  attached 
to  the  end  if  of  the  steam 
radius-rod.  The  end  h'  is 
movable  in  the  slot  of  the 
link,  being  carried  by  the 
hanger  m'ri,  which  is  at- 
tached at  ri  to  a  bell-crank 
iever  riop'  pivoted  at  <?,  the 
end  /'  being  moved  by  the 
governor. 

The  speed  of  the  engine 
is  kept  constant  by  the  move- 
ment of  the  steam-rod  which 
regulates  the.cut-off,  and  thus 
the  supply  of  steam  to  the 
cylinder.  The  rod  g' q'  is  the 
exhaust-rod,  and  is  attached 
permanently  to  one  point  of 
the  link  g' ,  and  moves  the  ex- 
haust-valves, of  which  there 

are  two  on  the  same  stem,  through  the  bell-crank  q'rs',  pivoted 
at  r,  and  the  rod  s't,  which  is  connected  at  /  to  the  valve- 
stem.  In  this  gear  the  radius  of  the  link  is  slightly  more 


96  VALVE-GEARS. 

than  the  length  of  the  steam  radius-rod.  The  effect  of  this 
is  that  as  the  radius-rod  is  lowered  or  approaches  the  cen- 
tre of  the  link  the  lead  on  one  end  is  decreased  and  the 
other  end  is  increased. 

It  will  be  noticed  that  in  this  figure  the  crank  and  eccen- 
tric are  nearly  on  the  same  line,  as  a  reverse  lever  is  used  in 
moving  the  valve.  That  they  are  not  exactly  together  is 
due  to  the  fact  that  the  end  i'  of  the  steam  radius-rod  h'i' 
does  not  move  in  the  line  aw,  but  along  the  line  af,  which  is 
drawn  through  the  middle  of  the  arc  in  which  i'  moves. 
The  use  of  the  two  driving  arms  on  the  rock-shaft  j  is  to 
cause  the  valves  to  move  as  fast  as  possible  when  opening 
and  closing  the  ports,  and  the  arms  are  so  arranged  that  F 
is  moving  at  its  fastest  rate  in  the  direction  l'uf  when  the 
front  valve  is  opening  or  closing,  and  k'  is  arranged  similarly 
for  the  back  valve. 

As  a  first  approximation  the  Zeuner  diagram  is  close 
enough  for  practical  work,  but  for  exact  work  with  any  Fink 
motion  the  valve-motion  should  be  laid  down  full  size,  to 
determine  the  actual  points  of  admission  and  cut-off.  The 
advantage  of  the  Fink  gear  is  its  simplicity,  and  its  disad- 
vantage is  that  the  cut-off  in  the  two  ends  of  the  cylinder 
may  vary  greatly,  as  the  variation  of  the  actual  diagram 
for  the  Zeuner  diagram  is  greatest  about  the  point  of  cut- 
off. The  dimensions  taken  in  the  Porter-Allen  engine  are 
such  that  the  cut  off  points  are  practically  symmetrical  up 
to  half-strokes  at  the  expense  of  unequal  leads. 

QUESTIONS. 

107.  Sketch  an  Allen  link,  and  explain  how  it  works. 

108.  Deduce  the  equation  to  the  movement  of  the  valve. 

109.  Is  the  lead  with  the  Allen  link  constant  or  variable  ? 
no.  Sketch  a  Fink  motion,  and  explain  how  it  works, 
in.  What  should  be  the  radius  of  the  link,  and  why  ? 

112.  What  is  the  distance  from  the  centre  of  the  shaft  to 
the  centre  of  the  exhaust-port  ? 

113.  How  determine  the  point  of  suspension  of  the  link? 


t 
THE  ALLEN  AND  FINK  MOTIONS.  97 

114.  Deduce  the  equation  for  the  movement  of  the  valve. 

115.  Explain  the  method  of  drawing  the  valve-diagram. 

116.  If  the  radius-rod  is  secured  at  one  point  in  the  link, 
what  is  the  value  of  x  ? 

117.  How  determine  the  curve  in  which  the  end  of  the 
hanger  moves? 

1 1 8.  How  should  the  eccentric  be  set? 

119.  Explain  the  method  of  designing  a  Fink  motion. 

1 20.  Sketch  and  explain  the  Porter-Allen  motion. 

121.  Why  is  the  eccentric  set  as  shown? 

122.  What  is  the  result  of  changing  the  radius  of  the 
link? 

PROBLEMS. 

46.  In  a  Fink  motion  r  =  i  inch,  a  =  6  inches,  b  =  o,  c  = 
10  inches,  radius-rod  40  inches,  draw  the  valve-diagram  for 
u  —  4,  8,  and  10  inches.     The  crank  being  20  inches  and  the 
connecting-rod   50,   where  is  the  point  of  cut-off  on   both 
strokes  for  each  value  of  u,  the  lead  for  u  =  10  being  y1^  inch  ? 

47.  Replace  the  Stephenson's  link  of  problem  40  by  a 
Fink  motion  which  will  give  equal  cut-off  at  .5  stroke,  mak- 
ing b  =  o  and  a  =  4!  inches. 


CHAPTER  XII. 

SHAFT  REGULATION. 

84.  Throttling  Governors. — With  a  single  eccentric  con- 
nected directly  to  a  common  slide-valve  the  point  of  cut-off 
is  fixed.     If   the  steam-pressure  is  constant,  the  amount  of 
work  done  by  an  engine  of  this  kind  will  vary  directly  with 
the  number  of  revolutions.     The  speed  of  the  engine  would 
be  consequently  varying  with  the  load.     To  overcome  this, 
some  device  must  be  used  to  keep  the  speed  constant.     With 
the  single  eccentric  and  ordinary  slide-valve,  the  pressure 
of  the  steam  must  be  modified  to  suit  the  work  to  be  done. 
This  is  usually  accomplished    by  means  of  some  form   of 
throttling  governor. 

It  is  generally  believed,  and  is  often  correct,  that  the 
use  of  the  throttling  governor  is  less  economical  than  some 
method  of  governing  which  regulates  the  time  during 
which  steam  is  admitted  to  the  cylinder,  or  in  other  words 
regulates  the  point  of  cut-off.  There  are  man)7  types  of  these 
governors,  one  of  which,  the  Porter-Allen,  acts  by  moving 
the  radius-rod  in  the  link,  and  thus  alters  the  point  of  cut-off. 

85.  Changing   Angular   Advance. — An   examination   of 
Fig.  54  shows  that  if  we  can  change  the  angle  of  advance  of 
the  eccentric  we  also  change  the  cut-off,  or  the  engine  would 
be  self-regulating  to  a  certain  extent.     But  the  lead  would 
also  change  in  a  manner  not  at  all  desirable.     As  lead  is 
only  given  that  the  piston  may  feel  the  full  pressure  of  the 
steam  at  the  beginning  of  the  stroke,  too  much  lead  would 
tend  to  stop  the  engine  before  it  reaches  the  dead-point,  and 
too  little  would  make  the  piston  move  under  less  than  the 
full  pressure   through   part  of  the  stroke,  the  consequence 

98 


SHAFT  REGULATION. 


99 


being  that  the  engine  would  be  doing  less  work  than  it  is 
capable  of  doing. 

86.  Changing  the  Eccentricity. — If  we  change  the  throw 
of  the  eccentric,  leaving  the  angle  of  advance  unchanged, 
we  would  find  the  same  trouble,  but  in  the  opposite  direc- 


FIG.  54. 

tion.  That  is,  when  the  angle  of  advance  only  is  changed, 
decreased  cut-off  means  more  lead.  When  the  eccentricity 
only  is  changed,  increased  cut-off  means  more  lead. 

87.  Changing  Eccentricity  and  Angular  Advance.— A 
combination  should  give  variable  cut-off  with  constant  lead. 
All  the  single-valve  automatic  engines  attempt  to  produce 
this  effect. 

In  Fig.  55,  if  ab  is  the  crank  and  ac  the  eccentric  when 
the  cut-off  is  greatest,  it  is  evident  that  gac  is  the  angle  of 
advance.  If  now  the  eccentric-sheave  be  made  so  that, 
without  changing  the  position  of  the  crank,  it  can  be  moved 
vertically  downwards,  the  slot  allowing  the  sheave  to  move, 
when  the  point  c  reaches  d  the  angle  of  advance  has  in- 
creased to  gad,  and  the  eccentricity  has  become  ad.  The 
positions  ac  and  ad  are  the  extreme  positions  possible,  and 
the  cut-off  can  vary  between  the  limits  set  by  these  eccen- 
tric positions.  Now,  if  when  the  engine  cuts  off  shortest 


1 00  VA L  VE- GEARS. 

the  power  developed  is  not  sufficient  to  run  the  engine  un- 
loaded, the  engine  will   regulate   between  a  full  load  and 


FIG.  55- 

unloaded  if  proper  mechanism  is  used  to  move  the  eccen- 
tric, as  shown  in  Fig.  55. 

88.  Erie  Governor. — Many  devices  are  used  to  accom- 
plish this  purpose,  one  of  the  simplest  being  that  shown  in 
Fig.  56,  which  represents  the  governor  used  on  the  engines 
built  by  the  Erie  City  Iron  Works.  The  shaft  carries,  per- 
manently attached  to  it,  a  frame  cc,  to  which  the  weights 


FIG.  56. 

a  and  a  are  attached.     Connected  to  these  weights  by  the 
bell-crank  arms  bb  is  the  eccentric-sheave,  which  is  slotted 


SHAFT  REGULATION.  IOI 

as  shown  in  Fig.  55.  As  the  speed  of  the  engine  increases 
above  the  normal,  the  balls  fly  out,  the  eccentric  moves 
across  the  shaft,  the  angle  of  the  advance  is  increased  the 
throw  of  the  eccentric  is  diminished,  and  the  cut-off  is 
earlier.  The  spring  at  the  bottom  of  the  figure  is  to  bring 
the  eccentric  back  to  its  position  of  greatest  throw  when 
the  speed  decreases  or  the  engine  is  stopped.  The  valve 
used  is  the  ordinary  piston-valve  directly  connected  with 
the  eccentric-rod. 

89.  Armington  and  Sims. — Another  method  of  produc- 
ing the  same  result  is  that  used  by  the  makers  of  the  Arm- 
ington and  Sims  engine.  The  apparatus  used  is  more 
complicated  and  the  results  are  no  better  as  far  as  the  valve- 
motion  is  concerned.  [This  discussion  does  not  deal  with 
the  adaptability  of  any  of  these  arrangements  for  quick 
governing,  nor  their  mechanical  excellence,  but  simply  with 
the  effect  of  changing  the  governor  position  on  the  valve.] 
Fig.  57  represents  this  governor  in  two  positions :  one,  A, 
showing  its  position  when  the  valve  has  its  least  travel ;  and 
one,  B,  with  the  greatest  travel.  In  the  figure  will  be  seen 
practically  two  eccentrics,  the  inner  one,  s,  loose  on  the 
shaft,  the  outer  one,/,  being  loose  on  the  inner  one.  The 
inner  one  carries  two  ears, /and  Ji,  which  are  attached  by 
links/'  and  hb  to  the  weights  de  and  ac. 

The  outer  eccentric,  /,  is  attached  to  one  weight  ac  by 
the  link  eg  attached  at  g.  The  fly-wheel  is  keyed  to  the 
shaft,  and  to  the  arms  at  a  and  d  are  pivoted  the  weights  of 
the  governor.  B  represents  the  governor  when  the  engine 
is  stopped,  the  combined  eccentrics  having  their  greatest 
throw,  of  is  the  position  of  the  crank  with  respect  to  the 
entire  arrangement.  The  engine  turns  in  the  direction  of 
the  arrow.  The  valve  is  a  piston-valve,  as  shown  in  Fig.  21  ; 
but  steam  is  taken  inside  and  the  exhaust  is  on  the  outer  side 
of  the  valve.  The  throw  of  the  eccentric  should  therefore 
be  in  the  same  position  as  though  an  ordinary  D-valve  was 
used,  moved  through  a  reverse-lever.  As  the  speed  of  the 
engine  increases,  the  weights  move  out  and  the  eccentrics 


102 


VALVE-GEARS. 


change  position,  so  that  when  they  occupy  the  position  shown 
in  A  the  valve  has  its  least  travel  and  the  cut-off  is  shortest. 

The     line    diagram 

I  shown    in  Fig.  58  will 

show  how  the  change 
in  travel  and  angu- 
lar advance  is  effect- 
ed, a  is  the  centre  of 
the  shaft  and  ag  the 
fixed  distance  from  the 
centre  of  the  shaft  to 
the  centre  of  the  joint 
carrying  the  weight 
abc  of  Fig.  57,  which 
weight  is  represented 
by  the  linens  gfe  in  Fig. 
58.  ab  is  the  eccen- 
tricity of  the  inner 
eccentric ;  ha  is  a  con- 
tinuation of  this  line, 
and  hf  is  a  link  con- 
necting the  point  h  to 
the  weight  gfe;  be  is 
the  eccentricity  of  the 
outer  eccentric ;  db  is 
connected  at  a  fixed 
angle  to  be,  and  d  and  e 
are  connected  by  the 
link  dc. 

As  the  speed  of  the 
engine  increases  efg 
moves  around  g  away 
from  ag,  h  approaches 
ag,  as  does  b.  At  the 
same  time  d  is  drawn 
downwards  and  c 
moves  still  closer  to 


SHAFT  REGULATION. 


I03 


ag,  until  when  at  its  greatest  speed  c  is  on  ag  produced,  and 
the  valve  has  its  least  travel. 

It  is  evident  that,  for  any  position  of  the  arrangement, 
ac  is  the  throw  of  the  combined  eccentrics,  and  the  angle 
kac  is  the  angle  of  advance.  In  order  that  the  lead  may  be 
constant,  the  point  c  should  move  at  right  angles  to  tf^as  the 
speed  changes.  The  figure  represents  the  parts  at  their 
greatest  and  least  cut-off. 


FIG.  58. 

90.  Ball. — The  cut-off  effected  by  the  Ball  engine  regu- 
lator is  practically  the  same  as  in  the  two  engines  just 
described.  Fig.  59  represents  the  regulator.  The  fly-wheel 
hub/  is  keyed  to  the  shaft  a.  The  outer  end  of  the  hub 
carries  an  eccentric  b  secured  by  the  four  bolts  q,  q.  c  is  the 
eccentric-strap  which  carries  the  plate  d,  to  which  is  attached 
a  pin  e  which  drives  the  eccentric-rod.  To  two  opposite 
arms  of  the  fly-wheel  at  ftf  the  weights/,/  are  pivoted, 
each  of  which  is  connected  to  the  eccentric-strap  c  by  the 
rods  k,  k.  On  the  line  ss  springs  are  attached  to  the  arm 
carrying  the  weights  and  to  the  rim  of  the  fly-wheel,  which 
act  against  the  centrifugal  force  tending  to  force  the  weights 
/,/  outward.  An  auxiliary  spring  connects  m  with  the  piston 
of  the  dash-pot  shown  at  o.  As  the  speed  of  the  engine 
increases  above  the  normal  the  weights  /,/  move  outward 


IO4 


VALVE-GEARS. 


and  turn  the  strap  c  on  the  eccentric,  causing  the  pin  e  to 
travel  in  an  arc  about  v,  the  centre  of  the  eccentric.     This 


FIG.  59. 


SHAFT  R 'EGO 'LA  TION. 


105 


varies  the  distance  from  e  to  <?,  or  the  eccentricity,  and  also 
varies  the  angular  advance.  In  the  two  regulators  before 
described,  the  eccentric  moved  at  right  angles  to  the  crank. 
In  this  case  the  eccentric  moves  about  a  fixed  point  v,  and 
the  change  in  distribution  of  steam  is  practically  the  same 
that  takes  place  in  a  Stephenson's  link  with  open  rods.  The 
valve  in  the  Ball  engine  takes  steam  inside,  so  that  the  eccen- 
trie  should  follow  the  crank  by  an  angle  equal  to  90°  less  the 
angle  of  advance.  The  lead  therefore  becomes  greater  as 
the  cut-off  is  shorter,  and  the  point  v  must  be  so  chosen  that 
the  lead  is  sufficient  when  loaded,  and  not  excessive  when 
running  light. 

91.  The  Valve  used  on  this  engine  is  different  from  any 
of  those  of  which  sketches  have  been  given.  The  valve  is 
on  the  side  of  the  cylinder,  and  Fig.  60  shows  a  vertical 


FIG.  60. 


section  through  the  valve  and  ports.  Steam  passes  from  the 
steam-pipe  a  into  the  interior  b  of  the  valve.  The  valve 
consists  of  the  parts  c  and  d,  which  slide  freely  in  each  other, 
but  without  allowing  steam  to  escape  into  the  space  h.  The 
upper  and  lower  end  of  the  pieces  c  and  d,  ii  and  jj,  are  ex- 


106  VALVE-GEARS. 

tended  to  make  the  valve-face  covering  the  rectangular  ports 
/,  /,  ey  and  e.  These  ports  lead  horizontally  into  the  cylinder, 
and  the  valve  is  double-ported.  Exhaust  takes  place  outside 
the  valve  into  the  space  h,  and  then  through  the  ports  g,  g 
in  the  bottom  of  the  chest  to  the  exhaust-pipe.  The  area 
exposed  to  the  steam-pressure  is  only  enough  to  insure  the 
valve  remaining  on  its  seat. 

QUESTIONS. 

123.  How  is  the  speed  of  an  engine  regulated  when  a 
plain  slide-valve  driven  by  a  single  eccentric  is  used? 

124.  How  is  the   speed  regulated   in   the    Porter-Allen 
engine? 

125.  Is  it  feasible  to  regulate  the  cut-off  by  changing  the 
angular  advance  only  ?     Why  ? 

126.  How  can  the  cut-off  be  regulated  by  changing  both 
the  angular  advance  and  the  eccentricity  ? 

127.  What  is  the  mechanism  used   on  the  Erie  engine? 
Is  the  variable  cut-off  the  same  as  would  be  obtained  by  a 
Stephenson's  link  ? 

128.  How  is  the  cut-off  varied  on  the  Armington  and  Sims 
engine,  and  what  link-motion  would  give  the  same  variation? 

129.  Describe  the  valve  used  on  the  Ball  engine,  and  the 
apparatus  used  for  changing  the  point  of  cut-off. 

130.  What   link-motion   is   the   equivalent   of    the    Ball 
motion? 

PROBLEMS. 

48.  Given    stroke    24  inches,  connecting-rod  60   inches, 
maximum   cut-off  at  f  stroke,  lead  \  inch,  port-opening  \\ 
inches.     What  must  be  the  movement  of  the  eccentric  to 
vary  the  cut-off  from  y3^  to  £  stroke  ? 

49.  Given  the  lap  \  inch,  lead  -fe  inch,  the, cut-off  to  vary 
from  i  to  f  stroke.     What  should  be  the  movement  of  the 
eccentric  ? 


CHAPTER  XIII. 
RADIAL  GEARS— HACKWORTH'S. 

92.  Radial  Gears. — There  is  another  class  of  valve-gears 
which  are  used  either  for  reversing  or  for  changing  the  point 
of  cut-off,  which  we  will  now  take  up,  and  which  are  called 
radial  gears.     A  radial  valve-gear  is  one  in  which  the  motion 
of  the  valve  is  taken  from  some  point  in  a  vibrating  link,  a 
second  point  of  which  moves  in  a  closed  curve,  while  a  third 
point  moves  in  a  straight  line  or  open  curve.     A  subdivision 
might  be  made  of  simple  and  compound  gears,  ^  gear  being 
simple  when  the  closed  curve  is  a  circle,  while  in  a  compound 
gear  the  closed  curve  is  described  by  a  point  on  a  second 
vibrating  link  moving  according  to  a  similar  law,  and  whose 
motion  may  be  either  simple  or  compound. 

93.  Hackworth's  Gear. — Hackworth's  gear  is  the  oldest 
and  simplest  radial  gear,  and  Fig.  61  gives  a  line  diagram 


FIG.  61, 

of  this  motion,     db  is  the  crank;  ac  is  the  eccentric  set  180° 
in  advance  of  the  crank ;  fe  is  the  valve-stern,  and  de  the 

107 


io8 


VALVE-GE4RS. 


valve  connecting-rod  ;  ch  is  the  vibrating-piece,  from  the 
point  d  of  which  motion  is  taken.  One  end  c  moves  in  the 
closed  curve  of  the  eccentric-circle,  while  the  other  end  h 
moves  in  a  straight  line  kg,  whose  direction  is  determined 
by  the  angle  of  inclination  a  which  kg  makes  with  the  line  of 
centres  ag.  When  the  piece  /^occupies  the  position  shown, 
an  ordinary  D-slide  taking  steam  outside  being  used,  the 
engine  turns  in  the  direction  of  the  arrow,  and  when  it  is  in 
the  position  h'g  the  engine  turns  in  the  opposite  direction. 

94.  Constant  Lead. — In  Fig.  62,  when  the  crank  is  on 
either  dead-point  a  or  b,  the  eccentric  being  180°  in  advance, 
the  length  of  the  eccentric-rod  be  or  ac  is  such  that  the  point 
c  coincides  with  the  centre  on  which  ie  turns.  Then  as  the 


FIG.  62. 

valve-rod  is  attached  at/,  the  distance  fg-=.fg-=.  lap  plus 
lead,  and  is  constant  for  both  ends  of  the  cylinder,  during 
both  forward  and  backward  motions,  and  for  all  grades  of 
expansion. 

95.  Movement  of  the  Valve. — Suppose  the  crank  to  have 
moved  through  the  angle  &>,  then  the  eccentric  has  moved 
to  the  position  oh  and  the  point  c  to  i  and  /to  k.  Draw  the 
lines  in,  km,  and  hp  at  right  angles  to  oc,  and  ir  parallel  to  oc. 
Then  km  is  the  movement  of  the  valve  if  the  angularity  of 
the  valve-rod  ks  is  neglected.  Let  be  =  ac  —  hi  =.ll,fc=  ki=  /, . 
Call  ico  =  a,  km  =  x,  and  oh  =  r.  Then 

x  =  km  =  kq  -f-  qm  —  kq  +  in, 


RADIAL   GEARS—  HACKWORTPTS.  109 

But 


Now  en  =  op  —  r  sin  <*?  approximately,  and  in  =  en  tan  a  — 
r  sin  GL>  tan  a,  and  r-#  =  hp  —  in  =  r  cos  GO  —  r  sin  6?  tan  <*  ;  or 
substituting,  we  have 

kq  =  -f  (r  cos  oo  —  r  sin  GO  tan  <*) 
A 

and 

#  =  7  (r  cos  oo  —  r  sin  G^>  tan  <*)  -f-  ^  sin  GJ  tan  a  ; 

^i 

or 

x  =  j  r  cos  oo  -[-  sin  &/  r  tan  a  —  ~  r  tan  « 

/ 


-  /.        \  . 

7  —  tan  «r  I  sm 


=  A  cos  GL>  -|-  .5  sin  60, 
which  is  the  equation  to  the  Zeuner  diagram  in  which 

A  =  l-f  r       and       B  =  r^-^]  tan  a. 
96.  The   Valve-diagram.  —  As    most   engines   to  which 


FIG.  63. 

these  gears  are  applied  are  vertical,  the  diagrams  are  drawn 
as  though  one  dead-point  was  at  the  top  of  the  figure  and 


1 1 0  VAL  VE-GEARS. 

one  at  the  bottom.     In  Fig.  63  is  drawn  the  diagram  for  this 
motion,  in  which 


/9r  r  //,  -  /' 

oa  =  -       and     av  =  -  -   tan  #. 


From  these  equations  we  see  that  whatever  the  value  of  ar, 
the  centre  of  the  valve-circle  is  on  the  line  ab.     Now 


ab      /,  -  /2 

tan  #0£  =  —  =  — -. —  tan  a. ; 
oa          / 


or  as  oba  would  be  the  equivalent  of  the  angle  of  advance  of 
a  single  eccentric,  to  produce  the  same  distribution  of  steam 


tan  d  =  77 rf . 

(/,  —  /2)  tan  a 


For  any  value  of  #,  therefore,  the  corresponding  valve- 
circles  can  be  drawn.  With  o  as  a  centre  and  the  lap  od  as 
a  radius  draw  the  lap-circle  def.  The  port  opens  when  the 
crank  is  at  od,  the  lead  is  ec,  and  the  cut-off  takes  place  when 
the  crank  reaches  of. 

97.  To  Design  the  Gear. — To  design  a  Hackworth 
motion,  suppose  we  have  /:,  /2,  the  lap,  and  the  lead  given, 
to  determine  the  value  of  r  and  of  a  for  a  given  cut-otf.  In 
Fig.  64  draw  oa  and  ch  at  right  angles  to  each  other.  From 
o  lay  off  oe  —  lap  and  ef  =  lead.  Draw  oi  to  represent  the 
position  of  the  crank  at  the  point  of  cut-off.  Draw  the  lap- 
circle  ehi.  The  valve-circle  must  pass  through  f,  0,  and  i. 
Finding  the  centre  k,  and  drawing  the  circle,  we  have 

/,  A  X  of 

of=jr      or     r  =  —r- 
t-i  *••> 


t 
RADIAL    GEARS— HACK  WORTH'S.  Ill 

and 

/.  —  /2  /2  tan  fok 

tan  fob  =  -L— —  tan  a     or     tan  a  =  — — , 

'a  *i  —  *« 

which  determine  the  other  details  of  the  gear. 

98.  Right-hand  Rotation. — For  convenience  in  dealing 
with  the  errors  of  this  diagram  we  will  use  the  term  "  right- 
hand  rotation"  to  mean  as  follows :  Assume  the  end  of  the 


FIG.  64. 

vibrating  link,  Figs.  61  and  62,  which  moves  in  a  closed 
curve  to  be  at  your  left,  the  end  which  moves  in  an  open 
curve  or  straight  line  to  be  on  your  right,  and  the  valve  to 
be  above  the  vibrating-link.  Rotation  in  the  direction  of 
the  movement  of  the  hands  of  a  watch  will  be  called  right- 
handed.  Then  in  Figs.  61  and  62  movement  in  the  direction 
of  the  arrows  is  right-hand  rotation. 

99.  Errors  of  the  Diagram — If  a  diagram  is  drawn 
according  to  the  formulas  deduced  above,  and  a  second  dia- 
gram is  made  showing  the  actual  position  of  the  valve  from 
a  scale-drawing  of  the  motion,  it  will  be  found  that  there  is 
an  error  in  the  Zeuner  diagram  as  applied  to  this  gear.  It 
will  be  found  that  the  actual  diagram  approaches  more 
nearly  the  theoretical  for  right-hand  rotation,  and  if  the 
diagram  is  laid  down  for  different  values  of  a,  the  smaller 


112 


VAL  VE-GEARS. 


this  angle  the  less  the  errors  of  the  diagram,  and  in  design- 
ing a  motion  of  this  kind  these  points  should  be  remembered. 


FIG.  65A. 


FIG.  656. 


Figs.  65  and  66  represent  the  Zeuner  and  actual  diagrams 
for  a  motion  of  the  same  proportions  as  Fig.  62 ;   Fig.  65 A 


FIG.  66A. 


FIG.  66B. 


being  for  right-hand  rotation  and  tan  a  =  .75,  and  656  being 
for  right-hand  rotation  with  tan  a  —  .4.  Fig.  66  represents 
corresponding  conditions  with  left-hand  rotation. 


t 

RADIAL    GEARS— HACK  WORTH'S.  11$ 

100.  Port-opening. — While  there  is  a  difference  in  the 
amount  of  port-opening  and  cut-off,  with  motion  in  either 
direction,  for  the  two  ends  of  a  cylinder,  the  difference  is 
greater  with  left-hand  rotation,  and  might  become  serious  in 
a  badly  designed  gear. 

101.  Connecting  up  a  Hackworth  Gear. — In  a  horizon- 
tal engine  advantage  could  be  taken  of  this  by  making  the 
more  rapid  valve-motion  at  that  end  of  the  cylinder  at  which, 
owing  to  the   angularity  of    the  connecting-rod,  the    more 
rapid  motion  of  the  piston  itself  occurs.     In  vertical  engines 
designed  to  run  left-handed  it  is  well  to  introduce  a  rocker- 
arm,  thus  reversing  the  motion  and  giving  the  wider  opening 
and  retarded  cut-off  on  the  up-stroke.     If  desired,  the  eccen- 
tric can  be  changed   180°,  or  the  swinging  link  attached  to 
the  crank  directly.     The  point  i  would  then  travel  on  the 
part  ce  instead  of  ci,  Fig.  62,  if  the  engine  turned  in  the  same 
direction,  and  a  rocker  would  have  to  be  used  or  steam  taken 
inside  the  valve. 

102.  Attaching    Valve-stem    outside. — Sometimes,    in- 
stead of  attaching  the  valve-rod  at  k  between  h  and  tt  as  in 
Fig.  62,   it  is  attached    at   a   point  outside,  as  at  k' .     Our 
reasoning  still  holds  good ;  but  the  /„  now  changes  sign,  and 
the  equation  to  the  movement  of  the  valve  is 


x  =      r  cos  GO  -|-  r(— — -)  tan  a  sin  GO. 


The  sliding  of  the  end  i  of  the  vibrating-link  along  the 
guide  ie  is  liable  to  excessive  friction  as  the  angle  a  increases, 
and  this  led  to  the  other  types  of  simple  radial  gear. 

103.  Equalizing  Port-opening. — By  the  use  of  an  equal- 
izing lever  it  is  possible  to  make  the  port-opening  on  both 
ends  of  the  cylinder  the  same  for  any  given  cut-off,  and  if 
desired  the  cut-off  in  the  two  ends  of  the  cylinder  can  be 
made  exactly  equal. 

Fig.  67  represents  the  path  of  that  point  of  the  radius-rod 


114 


VAL  VE-GEARS. 


FIG.  67. 


to  which  the  valve-stem  or  valve 
connecting-rod  is  attached,  for 
one  value  of  a  for  which  the 
port-opening  is  to  be  equalized. 
a  and  b  are  the  positions  cor- 
responding to  the  dead-points; 
ac  is  therefore  equal  to  be  or 
to  the  lap  plus  the  lead,  mn 
being  the  centre-line  of  the 
movement.  Make  kc  —  ch  =  the 
lap.  When  the  crank  is  in  such 
a  position  that  the  end  of  the 
radius-rod  is  at  /,  the  valve  is 
just  opening.  At  i  the  valve  is 
just  closing  the  same  port.  At 
f  the  valve  is  just  opening  to 
steam  on  the  other  end,  and  at 
g  cut-off  takes  place. 

With  g  and  f  as  centres  and 
the  length  of  the  valve  connect- 
ing-rod, ks  of  Fig.  62,  as  a  radius, 
describe  arcs  intersecting  at  o, 
and  from  i  and  j  describe  arcs 
intersecting  at  /.  Bisect  op  by 
the  line  rs.  With  /  and  q  the 
extreme  positions,  draw  indefi- 
nite arcs  /  and  u  with  the  same 
radius.  Select  a  point  on  rs 
as  a  centre,  such  that  if  an  arc 
with  sp  as  a  radius  is  drawn,// 
is  equal  toou.  Draw  sv  perpen- 
dicular to  the  direction  of  move- 
ment of  the  valve,  vsr  is  then 
the  angle  which  the  arms  of  the 
equalizing  lever  should  make 
with  each  other.  The  length  of 
the  arms  should  be  such  that 


RADIAL    GEARS— HACK  WORTH'S.  115 

while  the  end  of  the  arm  rs  travels  from  o  to  p,  the  end  of 
the  arm  sv  travels  twice  the  lap. 

104.  Equalizing  the  Cut-off. — The  cut-off  can  be  equal- 
ized by  determining  the  points  o  and  p  from  the  points 
actually  occupied  by  the  lower  end  of  the  valve  connecting-- 
rod when  steam  is  admitted  and  cut-off  at  the  required  points 
instead  of  from  the  points  j  and  i  and  g  and  /,  as  in  the  last 
article.  While  this  method  will  make  the  port-opening  and 
cut-off  for  one  grade  of  expansion  exactly  equal,  it  will  make 
them  more  nearly  equal  for  all  grades  than  if  no  equalizing 
lever  is  used. 

QUESTIONS. 

131.  What  are  radial  gears?  and  what  is  the  distinction 
made  between  simple  and  compound  gears? 

132.  Sketch  and  describe  the  Hackworth  gear. 

133.  How  does  the  lead  vary  with  different  values  of  a? 

134.  Deduce  the  equation  to  the  movement  of  the  valve, 
and   show  how  to  draw  the  valve-diagram   for  a  vertical 
engine. 

135.  What  data  is  required  and   how  proceed  to  deter- 
mine the  other  parts  of  the  gear? 

136.  Explain  the  distinction  made  between  right-  and  left- 
hand   rotation.     If  the  closed  curve  is.  on  your  right,  the 
open  curve  on  your  left,  and  the  valve  below  the  vibrating- 
link,  is  motion  with  the  hands  of  the  watch  right-  or  left- 
handed? 

137.  How  does  the  port-opening  vary  with  the  Hack- 
worth  gear?  and  is  the  variation  greater  for  right- or  for 
left-hand  rotation? 

138.  How   should    a   horizontal   engine    be    run   with   a 
Hackworth  gear ?     A  vertical?     Why? 

139.  What  effect  has  attaching  the  valve  connecting-rod 
to  an  extension  of  the  vibrating-link  ? 

140.  How  can  the  port-opening  be  equalized? 

141.  How  can   the  cut-off  be  equalized?   and  can  it  be 
done  for  several  points  of  cut-off  ? 


Il6  VALVE-GEARS. 


PROBLEMS. 

50.  Draw  the  Zeuner  diagram  for  a  Hackworth  gear  hav- 
ing r  —  2.5  inches,  /x  =  23^-  inches,  /2 '=  —  i6f  inches,  a  =  8^ 
and  25°.     Lap  =  ij  inches.     Determine  the  point  of  cut-off. 

51.  Given  the  maximum  cut-off  at  .7  stroke,  a  to  be  not 
over  24°.    Angle  of  lead  8°,  lap  \\  inches,  lead  f  inch,  centre 
of  shaft  to  centre  of  valve  connecting-rod  at  right  angles  to 
stroke  of  piston  40  inches,  the  valve  connecting-rod  to  be 
attached  outside,  find  the  value  of  /x ,  /„ ,  and  r. 

52.  Given /j  —  2 /finches,  /,  =  —  21  inches,  r=  3T37  inches, 
and  lap  =  2f  inches.     Required  the  value  of  a  for  cutting 
off  at  half-stroke,  the  angle  of  lead  being  6°. 

53.  With  the  data  of  Problem  52  and  the  length  of  the 
valve  connecting-rod  as  60  inches,  determine  the  equalizing 
lever  which  will  make  the  port-opening  the  same  at  the  two 
ends  of  the  cylinder  and  the  cut-off  at  half-stroke. 

54.  If  the  crank  is  16  inches  and  the  connecting-rod  64 
inches,  determine  an  equalizing  lever  which  will  admit  steam 
6°  before  the  beginning  of  each  stroke,  will  cutoff  at  exactly 
half-stroke  on  both  strokes,   and   will  give  the  same  port- 
opening  on  both  ends,  the  other  data  being  as  in  Problem  52. 


CHAPTER  XIV. 
RADIAL  GEARS— MARSHALL,  ANGSTROM,  AND  JOY. 

105.  Marshall's  Gear. — Fig.  68  is  a  line  diagram  of  a 
Marshall  gear,  which  is  more  commonly  used  than  any  other 
form  of  simple  radial  gear.  As  in  Fig.  61,  ab  is  the  crank, 
ac  the  eccentric,  ch  the  vibrating-link  or  radius-rod,  de  the 
valve  connecting-rod,  and  ef  is  the  valve-stem,  i  is  a  fixed 
point  on  the  frame  to  which  a  radius-rod  ig  is  attached, 
and  for  any  one  grade  of  expansion  and  direction  of  running 
ig  is  fixed  in  position,  and  g  is  therefore  practically  a  fixed 
point. 

To  g  is  attached  a  link  gh  =  gi  and  to  h  is  attached  the 
end  of  the  vibrating-link  ch.  h  therefore  travels  in  the  arc 


a 


FIG.  68. 

of  a  circle  jih',  which  replaces  the  straight  line  of  the  Hack- 
worth  motion. 

A  comparison  of  Figs.  68  and  61  will  show  that  the  same 
equation  will  represent  the  movement  of  the  valve  if  we  call 
a  the  angle  made  by  the  tangent  of  the  arc  jh  at  the  point  i 
with  the  centre-line  ia. 

106.  Errors  of  the  Zeuner  Diagram. — Replacing  the 
straight  line  by  the  arc,  brings  another  error  in  the  diagram, 

117 


n8 


VALVE-GEARS. 


which,  as  will  be  seen  from  Figs.  69  and  70,  is  quite  an  im 
portant  one.     Both  figures  show  the  Zeuner  diagram  in  the 


FIG.  69. 

full  lines  and  the  actual  movement  of  the  valve  in  broken 
lines  for  a  Marshall  gear  of  the  dimensions  given  in  Fig.  68. 
Fig.  69  shows  the  diagrams  for  right-hand  rotation,  and 


FIG.  70. 

Fig.  70  for  left-hand.     In  each  figure  A  is  for  tan  a  =  .75, 
and  B  is  for  tan  a  =  .4. 


t 
RADIAL   GEARS— MARSHALL,   ANGSTROM,   AND  JOY.     119 

An  examination  of  these  figures  will  show  that  for  left- 
hand  rotation  the  points  of  cut-off  and  port-openings  for  cor- 
responding positions  of  the  up  and  down  strokes  are  more 
nearly  equal.  Horizontal  engines  should  therefore  be  made 
to  run  in  this  way.  For  right-hand  rotation  the  port-open- 
ings for  corresponding  positions  on  the  up  and  down  strokes 
are  more  unequal,  and  greater  port-openings  and  more 
retarded  cut-off  are  given  on  the  up-strokes.  Vertical 
engines  should  therefore  be  made  to  run  in  this  direction, 
or,  as  is  more  commonly  done,  the  eccentric  could  be  set 
.with  the  crank,  and  the  valve  driven  from  ch  extended. 

It  is  possible  to  work  out  a  formula  which  gives  more 
exactly  the  actual  movement  of  the  valve,  but  the  results  are 
too  unwieldy  for  practical  use. 

The  chief  recommendation  of  this  gear  is  the  small 
number  of  working  parts,  and  the  excellent  distribution  of 
steam  which  can  be  obtained  by  a  proper  proportion  of  the 
parts. 

107.  Proportions  of  Gear. — The  dimensions  of  the  differ- 
ent parts  of  this  gear  affect  seriously  the  points  of  cut-off, 
etc. ;  and  the  following  are  given  by  Mr.  G.  A.  C.  Bremme, 
the  inventor,  as  good  proportions  for  the  various  parts : 

Radius-rod  and  arm  gi  (Fig.  68)  =  6ac  ; 

Eccentric-rod,  ch  _=  6ac ; 

Lead  arm  hd  =  —  4.5^; 

Lap  on  steam  edges  (both  sides  equal)  =  .6ac. 

The  angle  a  should  never  exceed  25°. 

The  problems  give  the  principal  dimensions  of  gears 
actually  constructed  which  differ  very  materially  from  the 
proportions  given  above. 

108.  Designing. — The  method   of   designing  the   parts, 
laying  down  the  valve-diagram,  and  determining  an  equaliz- 
ing lever  are  the  same  as  already  set  forth  under  the  Hack- 
worth  gear. 


I2O 


VAL  VE-GEARS. 


109.  Angstrom's  Gear. — Angstrom's  gear  is  a  modifica- 
tion of  Hackworth's,  and  within  certain  limits  is  the  exact 
equivalent  of  it.  In  Fig.  71  the  end  h  of  the  vibrating-lever, 


FIG.  71. 

instead  of  being  carried  on  a  guide,  is  attached  to  a  point  h 
in  a  rod  kl,  the  ends  k  and  /  swinging  around  the  points  g 
andy  by  the  links  gl  and  jk,  the  points^  and  j  being  fixed 


FIG.  72. 

for  any  one  point  of  cut-off.  The  whole  arrangement  forms 
a  parallel  motion.  For  a  short  distance  on  either  side  of  i 
the  motion  of  h  is  at  a  fixed  angle  tojg  or  to  at,  and  when  b 
is  not  allowed  to  travel  beyond  this  limit  the  formulae  and 
diagrams  belonging  to  the  Hackworth  motion  fit  this  gear. 
If,  however,  this  limit  is  passed  another  error  is  introduced, 


f 
RADIAL   GEARS— MARSHALL,    ANGSTROM,   AND  JOY.     121 

and  its  amount  will  depend  on  whether  the  parallel  motion 
is  made  as  in  Fig-.  71  or  72,  either  of  which  will  give  motion 
at  the  same  angle  a  to  ai. 

no.  The  Diagrams.— If  the  diagrams  are  drawn  for  this 
gear  as  for  the  Hackworth  and  Marshall,  it  will  be  found 
that  the  results  are  much  better  if  the  parallel  motion  is 
connected  as  shown  in  Fig.  71,  and  right-hand  rotation 
should  be  selected  as  giving  more  even  cut-off  and  greater 
regularity  of  opening.  This  gear  is  superior  to  Marshall's 
but  inferior  to  Hackworth's  on  the  points  mentioned,  but  it 
must  be  remembered  that  this  only  applies  when  the  limits 
of  the  parallel  motion  are  exceeded.  As  regards  construc- 
tion, it  is  inferior  to  Marshall's  in  that  it  has  more  links, 
and  consequently  less  rigidity,  while  it  has  the  advantage 
over  Hackworth's  of  dispensing  with  the  sliding  motion  of 
the  end  of  the  vibrating-lever. 

in.  Advantages  and  Disadvantages  of  Radial  Gears. 
— There  is  a  disadvantage  under  which  all  these  gears  labor. 
To  give  a  well-balanced  motion,  the  vibrating-lever  ch  must 
be  long ;  and  as  all  the  stress,  which  is  considerable  in  the 
case  of  an  unbalanced  valve  or  of  high  speeds,  comes  trans- 
versely on  it,  there  is  likely  to  be  great  vibration  and  danger 
of  breaking. 

The  general  advantages  which  are  characteristic  of  all 
forms  of  radial  gears  are  lightness,  compactness,  a  small 
number  of  moving  parts,  and  constant  lead.  There  are  sev- 
eral forms  of  compound  radial  gears,  but  the  discussion  of 
one  will  cover  the  ground  sufficiently. 

112.  The  Joy  Gear. — Fig.  73  is  a  line  sketch  of  a  Joy 
gear  as  applied  to  a  horizontal  engine,  ab  is  the  crank,  be 
the  connecting-rod.  To  a  point  d  in  the  connecting-rod  is 
attached  the  link  de,  the  end  e  of  which  swings  about  a  point 
/on  the  engine  frame,  the  points  e  and /being  connected  by 
a  link  ef.  ig  is  a  rod  having  its  lower  end  g  connected  to 
the  link  de\  at  its  upper  end  /  is  connected  to  the  valve-rod 
ij,  which  connects  with  the  valve-stem  at/.  The  point  h  of 
ig  is  guided  along  the  arc  of  a  circle  Ikm,  either  by  guides  or 


122 


VAL  VE-GEARS. 


by  swinging  around  a  centre.     This  arc  is  pivoted  at  k  so 
that  the  cut-off  can  be  changed  or  the  engine  reversed. 


FIG.  73. 

Remembering  our  definition  of  right-hand  rotation,  the 
direction  shown  by  the  arrow  in  Fig.  73  is  left-hand. 

The  reason  for  compounding  a  radial  gear  is  to  do  away 
with  the  eccentric.  The  figure  shows  the  exact  motion  of 
the  points  d,  g  and  i. 

If  the  end  of  hg  were  attached  directly  to  the  connecting- 
rod  at  d,  the  point  d  during  the  lower  half-revolution  of  the 
crank  would  describe  the  lower  half  of  a  curve  which  nearly 
coincides  with  a  circle  having  k  as  a  centre,  so  that  h  would 
have  little  motion  along  Im,  and  the  only  motion  given  to 
the  valve  would  be  that  due  to  the  oscillation  of  hi  about  k. 
By  causing  g  to  describe  the  irregular  oval,  the  travel  of  the 


RADIAL   GEARS— MARSHALL,   ANGSTROM,   AND  JOY.     12$ 

point  h  above  and  below  the  centre  of  suspension  k  of  the 
arc  Im  is  equal  and  approximately  symmetrical. 

113.  Movement  of  the  Valve. — When  the  engine  is  on 
either  dead-point,  call  0  the  angle  the  link  ed  makes  with  a 
vertical  line,  and  0  the  angle  kg  makes  with  the  vertical. 
Call  the  distance  de  =  /„,  dg  •=•  /4,  hg  •=.  llt  ki  =  /2.  Then 
/4  vers  0  =  /!  vers  0,  as  the  angularity  of  the  rods  should 
neutralize  each  other  in  a  well-designed  gear.  Also,  /4  sin 
0  +  ^1  sin  0  —  ^  —  radius  of  crank,  as  the  point  ^  should 
travel  symmetrically  under  k.  Also  /,  sin  6  —  R. 

In  designing,  the  most  convenient  assumption  is  that 
which  fixes  the  value  of  /4  and  I, .  The  distance  from  the 
centre  of  the  valve-stem  to  the  axis  of  the  cylinder  is  fixed, 
and  is  approximately  /2  +  A  —  /4 .  Having  assumed  these 
values,  we  have  /4(i  —  cos  B)  =  /,(i  —  cos  0), 

/4  sin  6  -f-  /,  sin  0  =  R     and     /3  sin  #  =  R. 

From  these  equations,  by  eliminating  6  and  0,  we  can  get  a 
value  of  /3  in  terms  of  /4 ,  /x ,  and  7?. 

In  the  Hackworth  gear  the  movement  of  the  valve  is 
composed  of  two  parts,  one  part  mq,  in  Fig.  62,  due  to  the 
movement  of  i  along  ice,  and  one  part  qk  due  to  the  turning 
of  ir  around  i.  In  the  Joy  gear,  Fig.  73,  the  point  d  moves 

vertically  a  distance  -rR  sin  GO  when  c  =  the  distance  cd  and 

b  =  cb,  and  d  is  a  distance  R  cos  &?  from  its  middle  position. 
g  has  moved  vertically  practically  the  same  distance  as  d,  or 

-R  sin  GO,  and  horizontally  is   3   .    -R  cos  00  from  its  middle 

position. 

If,  then,  instead  of  r  sin  GO  in  the  equation  to  the  Hack- 
worth  motion,  we  put  -=-R  sin  GO,  and  instead  of  r  cos  GO  we 

put  3  -7?  cos  CB?,  we  get  the  equation  to  the  movement  of 
the  valve. 


124 


VALVE-GEARS. 


As  i  is  beyond  h,  the  equation  for  the  Hackworth  mo- 
tion is 


x  =  j  r  cos  oo-\-r  (^—,  —  -  tan  a  J  sin  GO  • 
and  making  the  substitutions  noted  above,  we  have 

X  =  IL=^-RJ  cos  oo  +  R  (^±^  tan  «)  j  sin  »,. 

from  the  movement  of  the  valve  from  its  middle  position. 
The  co-ordinates  of  the  centre  of  the  valve-circles  are  hori- 

zontally l-*^Rj>  and  vertically  ^^(^y-/a  tan  a}. 

From  these  values  the  valve-circle  can  be  drawn,  and  the 
points  of  cut-off,  steam  admission,  etc.,  determined. 

114.  Errors  of  the  Zeuner  Diagram.  —  The  errors  in 
using  the  diagram  are  shown  in  Figs.  74  and  75,  where  the 


FIG.  74- 

arrows  indicate  the  movement  of  the  crank :  that  is,  Fig.  74 
is  for  left-hand  rotation  by  definition,  and  Fig.  75  is  for  right- 
hand  rotation,  A  being  drawn  for  tan  a  =  .75,  and  B  for 
tan  a  —  .4.  It  will  be  seen  that  the  lead  is  equal  on  both 
^nds  of  the  cylinder  for  motion  in  either  direction,  but  the 
cut-off  is  more  nearly  equal  for  right-hand  rotation  than  for 
left. 


RADIAL    GEARS— MARSHALL,    ANGSTROM,   AND  JOY.     12$ 


QUESTIONS. 

142.  Sketch  and  describe  a  Marshall  gear. 

143.  Deduce  the  equation  for  the  movement  of  the  valve 
with  the  Marshall  gear. 

144.  What  parts  of  the  valve  movement  are  most  affected 
by  the  errors  of  the  Zeuner  diagram  ? 

145.  How  should  a  horizontal  engine  run  with  a  Marshall 
gear?     A  vertical?     Why? 

146.  Explain  the  method  of  designing  a  Marshall  gear. 


FIG.  75. 

147.  Sketch  Angstrom's  gear,  and  what  is  the  objection 
to  it? 

148.  What  are  generally  the  advantages  and  disadvan- 
tages of  radial  gears? 

149.  Sketch  a  Joy  gear. 

150.  Determine  the  movement  of  the  valve  in  the  Joy 
gear. 

151.  Is  the  Zeuner  diagram  a  close  approximation  in  the 
case  of  a  Joy  gear? 

152.  If  the  smaller  opening  of  the  port  is  desired  on  the 
end  opposite  that  which  the  gear  as  connected  would  give 
it,  what  must  be  done  to  change  it? 


1 26  VAL  VE-GEARS. 


PROBLEMS. 

Problems  50,  51,  and  52  apply  equally  well  to  the  Mar- 
shall gear. 

55.  Solve   Problem  53  for  a   Marshall  gear,  having  the 
reversing-arm  and  radius-rod  each  18^  inches. 

56.  Solve  Problem  54  for  a  Marshall  gear  having  the  re- 
versing-arm i8£  inches. 

57.  In  a  Joy  gear  having  crank  8  inches,  £=13   inches, 
b  =  39  inches,  /,  —  16  inches,  lz  =  8  inches,  /3  =  24  inches, 
and  /4  —  8  inches,  the  lap  being  2  inches,  and  the  reversing- 
arm  and  radius-rod  12  inches,  find  the  value  of  a  for  cutting 
off  at  GO  —  90°,  the  angle  of  lead  being  8°. 

58.  Draw  a  diagram  for  the  Joy   gear  in    Problem  57, 
showing  the  error  of  the  Zeuner  diagram  for  cutting  off 
at  oo  ~  90°  on  both  strokes. 

59.  If  the  engine  is  to  cut-off  exactly  at  half-stroke  on 
both  ends,  what  must  be  the  lap  and  the  lead  on  each  end. 

60.  In  a   Marshall  gear   having  r  =  5J-  inches,  lap   i£", 
/j  ==  66,  and  /2  =  30,   the  reverse   lever  30  and   tan  a  =  .6. 
Find  the  points  of  cut-off  in  both  ends  of  the  cylinder  from 
the  Zeuner  diagram,  and  by  laying  down  the  gear,  if  the 
stroke  is  48  inches;  and  the  connecting-rod  100  inches. 


CHAPTER   XV. 
DOUBLE  VALVES— GRIDIRON   VALVE. 

115.  Kinds  of  Double  Valves. — We  have  already  seen 
that  in  the  case  of  a  single  valve  moved  by  an  eccentric, 
early  cut-off  makes  equally  early  admission,  and  affects  the 
exhaust  in  the  same  way  as  the  steam.    To  retain  the  proper 
opening  and  closing  of  the  exhaust  while  varying  the  cut-off, 
double  valves  are  used,  and  may  be  divided  into  two  general 
classes. 

In  the  first  class  the  second  valve  called  the  cut-off 
valve  moves  on  an  independent  valve-seat,  and  controls  the 
admission  of  steam  to  the  steam-chest  of  the  main  valve. 
By  cutting  off  the  admission  of  steam  to  the  main  valve- 
chest,  steam  is  prevented  from  entering  the  cylinder  whether 
the  main  valve  is  open  for  steam  or  not.  As  the  exhaust 
takes  place  under  the  main  valve,  no  change  is  made  in  the 
exhaust. 

In  the  second  class  the  cut-off  valve  moves  on  the  back 
of  the  main  valve,  and  produces  the  same  effect. 

116.  Gridiron  Valve. — The  first  class  is  represented  in 
Fig.  76,  which  is  a  sketch   of  a  Gonzenbach    or   gridiron 
valve,     a  is  the  main  valve,  b  is  the  valve-chest,  c  is  the  seat 
of  the  cut-off  valve,  and  d  the  cut-off  valve  itself. 

Steam  first  enters  the  space  e  above  the  cut-off  valve,  and 
when  this  valve  is  open  passes  through  the  ports  g  and  h, 
into  the  main  valve  chamber  b.  If  the  valve  a  opens,  steam 
then  passes  into  the  cylinder.  After  steam  is  cut  off  by  the 
valve  d,  the  steam  in  the  chest  b  and  in  the  cylinder  expands 
as  one  volume;  and  after  the  main  valve  closes,  the  steam  in  the 
cylinder  alone  expands.  Exhaust  takes  place  under  the  valve 

127 


128 


VALVE-GEARS. 


a  as  in  an  ordinary  slide.     The  valves  a  and  d  are  moved  by 
independent  eccentrics,   but  the  valves  never  occupy   the 


FIG.  76. 

relative  positions  shown  when  connected  to  their  eccentrics. 
The  small  diagram  shows  the  relative  position  of  the  crank 
and  the  two  eccentrics. 

117.  Polonceau  Valve. — A  valve  of  the  second  class  is 
shown  in  Fig.  77,  which  is  a  sketch  of  Polonceau's  valve. 
The  valves  are  in  their  middle  position  at  the  same  time — a 


FIG.  77. 

position  they  never  assume  if  connected  to  their  eccentrics, 
a  is  the  main  valve,  which  has  two  passage-ways  b  and  c,  the 


DOUBLE    VALVES— GRIDIRON    VALVE. 


129 


part  between  the  passage-ways  being  an  ordinary  D-slide, 
On  the  back  or  tops  of  the  main  valve  is  a  cut-off  valve  dr 
which  is  a  plain  flat  block.  Each  valve  is  moved  by  its  own 
eccentric. 

As  the  main  valve  moves  to  the  left  steam  passes  from 
the  chest  e  through  the  passage  3,  to  the  steam-passage  lead- 
ing to  the  right-hand  end  of  the  cylinder.  The  block  d  is- 
moving  at  the  same  time,  and  by  a  proper  setting  of  the 
eccentrics,  at  the  right  point  in  the  stroke  the  valve  d  covers 
the  passage  b,  and  steam  is  cut  off.  The  exhaust  here  as  in 
the  other  class  takes  place  independently  of  the  admission 
and  cut-off.  The  small  diagram  shows  the  relative  position 
of  the  crank  and  the  two  eccentrics. 

118.  Diagram  for  Gridiron  Valve.— To  determine  the 
point  of  cut-off,  etc.,  of  the  Gonzenbach  valve,  we  proceed 


FIG.  78. 

as  follows  :  In  Fig.  78  draw  the  diagram  for  the  main  valve 
as  for  any  single  valve.  As  the  exhaust  is  not  affected  by 
the  cut-off  arrangement,  we  will  omit  the  exhaust  lines  from 
the  diagram.  In  the  figure  A  is  the  admission  and  B  the 


J30 


VAL  VE-GEARS. 


cut-off  in  tne  right-hand  end  of  the  cylinder,  C  is  the  admis- 
sion, and  D  the  cut-off  for  the  left-hand  end. 

Suppose  the  cut-off  valve  to  be  moved  by  an  eccentric 

without  angular  advance.  Fig. 
79  will  then  represent  the  dis- 
tance the  cut-off  valve  has 
moved  from  its  central  position 
for  any  position  of  the  crank. 
If  the  width  of  the  port  h  is  e 
and  of  g  is  /,  it  is  evident  that 
if  the  valve  moves  away  from 
its  central  position  a  distance 

e+f 

— —  =  s,  the  edge  of  d  is  just 

closing  the  port  h.  If  in  Fig. 
79  we  draw  a  circle  efg  with  s 
as  a  radius,  the  cut-off  valve 
covers  the  port  while  the  crank 
is  passing  from  E  to  F. 

Referring  to  Fig.  78,  it  is 
evident  that  the  cut-off  valve  must  open  between  D  and  A 
in  order  that  the  steam  may  be  admitted  as  soon  as  the  main 
valve  opens.  It  must  close  between  A  and  B  to  cut  off 
before  the  main  valve  does,  and  it  must  not  open  again  until 
the  crank  passes  B,  otherwise  steam  would  be  admitted  twice 
during  one  stroke.  The  cut-off  valve  must  again  open  be- 
tween B  and  C  to  admit  steam  to  be  ready  for  the  next 
stroke. 

That  is,  in  a  combined  diagram  for  the  two  valves  E 
should  fall  between  A  and  B,  and  F  should  fall  between  B 
and  C,  and  this  generally  requires  that  the  angle  of  advance 
of  the  cut-off  valve  should  be  negative. 

119.  Combined  Diagram  for  both  Eccentrics. — Fig.  80 
represents  such  a  combined  diagram.  Steam  is  admitted 
through  the  cut-off  valve  at  F  and  the  main  valve  opens  at 
A.  Cut-off  by  the  cut-off  valve  takes  place  at  E  and  by  the 
main  valve  at  B.  The  cut-off  valve  opens  again  at  F,  and 


FIG.  79. 


DOUBLE    VALVES— GRIDIRON    VALVE.  131 

the  main  valve  to  the  other  end  of  the  cylinder  at  C.  As 
the  point  F  must  always  fall  between  B  and  Cy  the  limits  of 
cut-off  obtainable  by  this  gear  are  not  very  great. 

If  E  falls  on  R,  the  main  and  cut-off  valve  close  at  the 
same  time  ;  and  if  F  falls  on  C,  the  cut-off  takes  place  at  £lt 


FIG.  80. 


and  the  cut-off  can  therefore  be  changed  by  changing  the 
value  of  s  between  oe'  and  oB. 

120.  Limits  of  Cut-off. — In  the  diagram  we  have  drawn, 
the  centre  of  the  cut-off  valve-circle  falls  on  oB.  If  it  falls 
on  any  other  line,  as  oB',  the  limits  of  the  cut-off  would  be 
less.  For  if  ol'  is  such  a  value  of  s  that  it  cuts  the  cut-off 
valve-circle  on  oB,  the  cut-off  takes  place  on  oE^ ,  and  any 
greater  value  of  s  will  cause  the  valve  to  admit  steam  twice 


132  VALVE-GEARS. 

during  the  stroke.  The  range  of  cut-off  is  then  from  oEz  to 
oE^ .  This,  while  allowing  an  earlier  cut-off,  gives  less  range, 
and  at  the  same  time  makes  a  bad  distribution  of  steam  if 
a  later  cut-off  than  oE^  is  required. 

It  is  generally  better  to  draw  the  diagram  so  that  the 
centre  of  the  valve-circle  for  the  cut-off  valve  is  on  the  line 
of  cut-off  for  the  main  valve. 

121.  Width  of  Ports. — In  laying  down  the  diagram  all 
the  data  of  the  main  valve  are  given,  or  can  be  determined 
as  already  shown  for  a  plain  slide.     The  area  of  the  port 
through  the  seat  of  the  cut-off  valve  should  be  as  great  as 
and  preferably  a  little  greater  than  the  area  through  the 
main  valve-seat.     This  will  determine  the  value  of  e. 

The  opening  in  the  cut-off  valve  is  usually  greater  than 
this.  The  reason  for  this  is,  that  the  valve  may  be  opened 
and  closed  quickly,  or  for  the  same  reason  that  the  over- 
travel  is  given  to  an  ordinary  slide-valve.  Evidently  the 
port  begins  to  close  when  the  valve  has  travelled  a  distance 

-Ll  -  from  its  middle  position,  is  entirely  closed  when  x  —  s, 
begins  to  open  again  when  x  is  equal  to  s,  and  is  wide  open, 
again  when  x  = . 

122.  Angle  of  Advance. — Having   determined  e  and  /, 
suppose  in  Fig.  81  we  have  the  main  valve-circle  given,  cutting 
off  at  ob,  and  suppose  it  is  desired  with  the  cut-off  valve  to  cut 
off  steam  when  tne  crank  is  at  oe.     With  a  radius  equal  to 
s,  draw  the  circle  fgh.     Then,  in  order  that  the  cut-off  valve 
will  not  admit  steam  again  before  the  main  valve  cuts  off, 
the  valve-circle  for  the  cut-off  valve  must  pass  through  /,  <?, 
and  ^,  or  some  point  beyond  h  on  the  line  oh.     Making  it 
pass  through  h  gives  oi  for  the  throw  of  the  eccentric,  and 
got  for  the  angle  of  advance,  which  is  negative.     This  is  the 
least  travel  and  the  least  angle  of  advance  this  cut-off  eccen- 
tric can  have. 

The  greatest  angle  of  advance  and  travel  of  the  valve  can 


. 


DOUBLE    VALVES— GRIDIRON   VALVE.  133 

be  determined  by  causing  the  valve-circle  to  pass  through 
o,  f,  and  d,  the  admission  to  the  other  end  of  the  cylinder. 
If  the  engine  is  to  cut  off  permanently  at  one  point,  an  inter- 
mediate value  should  be  taken  to  make  sure  that  the  port  is 
well  open  at  the  admission  for  the  return  stroke. 

123.  Varying  Cut-off. — As  a  certain  range  of  cut-off  is 
possible  for  a  given  valve,  we  will  see  how  it  can  be  obtained, 


FIG.  81. 

and  the  engine  made  self-regulating  within  that  range.  An 
examination  of  Fig.  81  shows  that  if  the  angle  of  advance 
alone  is  increased  the  cut-off  is  later,  but  it  cannot  be  made 
less  than  that  shown  in  the  figure.  If  the  value  of  oi  alone 
is  increased  the  cut-off  is  earlier,  but  oi  can  only  be  increased 
until  the  cut-off  valve-circle  passes  through  d. 

If  the  angle  of  advance  then  is  increased  \.o  gob  and  the 
valve-circle  be  made  to  pass  through  h  and  o,  the  cut-off  will 
be  on  ob.  This  of  course  gives  the  latest  possible  cut-off. 
If  now  the  eccentricity  is  increased,  the  cut-off  is  earlier  and 


134  VAL  VE-  GEARS. 

earlier,  until,  when  the  valve-circle  passes  through  d,  it  is  the 
earliest  possible. 

With  a  given  eccentricity  the  earliest  cut-off  is  obtainable 
by  giving  it  such  an  angle  of  advance  that  the  valve-circle 
passes  through  h,  and  the  latest  by  making  it  pass  through 
d.  A  range  of  cut-off  equivalent  to  the  angle  hod  only  is 
obtainable,  and  this  small  limit  of  variation  is  the  reason 
why  this  cut-off  is  not  more  commonly  used. 

124.  Arrangement  used  for  Varying  Cut-off. — An  ar- 
rangement has  been  devised  by  which  the  travel  of  the  cut- 
off valve  can  be  changed,  thereby  changing  the  point  of 
cut-off.  In  Fig.  82  a  is  the  shaft,  ab  is  the  crank,  ac  the 


FIG.  82. 

main-valve  eccentric,  cd  its  eccentric-rod,  and  de  the  valve- 
stem,  fg  is  the  cut-off  valve-stem,  gh  a  radius-rod,  the  end  h 
being  connected  to  one  end  of  a  reverse-lever  hij  swinging 
around  i.  The  end  /  of  the  reverse-lever  is  connected  by 
the  eccentric-rod  jk  to  the  cut-off  eccentric  ak,  which  is  set 
in  the  position  shown  because  of  the  reverse-lever  being 
used  (the  cut-off  eccentric  having  negative  angular  advance). 
ih  is  an  arc  having  hg  as  a  radius.  By  moving  h  in  this 
arc  the  travel  of  the  valve  is  changed  in  the  proportion  of 
ih  to  if. 

The  part  of  ih  over  which  h  can  move  is  limited  by  the 
greatest  and  least  allowable  travel  from  the  diagram. 

125.  Width  of  Cut-off  Valve.— The  width  of  the  piece 
d  of  Fig.  76  is  yet  to  be  determined.  From  its  middle  posi- 
tion the  valve  moves  in  either  direction  a  distance  rl  =  the 
throw  of  the  cut-off  eccentric.  The  distance  from  the  edge 
u  to  /  should  be  greater  than  r^ ,  or  the  outer  edge  of  the 


DOUBLE    VALVES— GRIDIRON   VALVE.  135 

valve  sbould  never  uncover  the  port.  Therefore  the  width 
of  the  valve  with  a  single  port  should  be  greater  than 
2r  -4-  €. 

126.  Varying  Width  of  Block. — There  is  one  other  way 
in  which  this  valve  could  be  made  to  vary  the  cut-off,  and 
that  is  by  changing  the  value  of  s.  In  one  gear  this  is 
done  by  causing  the  cut-off  valve  to  slide  on  a  plate  which 
is  adjustable,  changing  the  value  of  e  and  the  point  of 
cut-off. 

QUESTIONS. 

153.  Why  are  double  valves  used  ? 

154.  Sketch  and  describe  the  two  classes  of  double  valves 
treated  of. 

155.  Explain  the  method  of  drawing  the  diagrams  for  a 
gridiron  valve. 

156.  What  limits  are  there  to  the  position  of  the  valve- 
diagram  for  the  cut-off  valve  ? 

157.  What  are  the  limits  of  cut-off  with  a  gridiron  valve? 

158.  What  fixes  the  width  of   port  in  the    cut-off  valve 
and  its  seat  ? 

159.  In  designing  a  gridiron  valve  what  considerations 
govern  the  fixing  of  the  angular  advance  ? 

160.  How  can  the  cut-off  be  varied,  and  what  arrange- 
ment  is  used  for  that  purpose? 

161.  How  determine  the  width  of  the  cut-off  valve  ? 

162.  Draw  the  valve-diagram  of  the  gridiron  valve  and 
show  the  effect  of  varying  the  width  of  the  port  in  the  cut- 
off valve. 

PROBLEMS. 

61.  Having  given  r  =  r^  =  2$-  inches,  tf  =  —  tf,  =  18°,  lap 
•|  inch,  e  =  i  inch,/=  ij  inches,  determine  the  point  of  cut- 
off and  the  variation  of  cut-off  which  would  be  possible  by 
changing  the  eccentricity  of  the  cut-off  eccentric. 

62.  The  main  valve  is  to  cut-off  at  f  stroke  #,  =  —  5°, 
maximum  r,  =  3  inches,  e  =  1.5  inches,  /=  2  inches,  what 


J  36  VAL  VE-GEARS. 

are  the  limits  of  rlt  and  between  what  values  of  co  can  the 
cut-off  be  varied  ? 

63.  The  main  valve  is  to  cut  off  at  f  stroke,  and  the  cut- 
off with  the  cut-off  valve  is  to  vary  from  J  to  f  stroke,  e  =  i£, 
/=  i|,  what  must  be  the  angular  ad vance  and  limiting  values 
oirj 


CHAPTER    XVI. 
RELATIVE  MOVEMENT- -POLONCEAU  GEAR. 

127.  One  Valve  on  the  Back  of  Another. — To  investi- 
gate a  valve-motion  of  the  second  class,  we  must  first  deter- 
mine the  relative  motion  of  two  valves,  each  moved  by  an  ec- 
centric, when  one  moves  on  the  back  of  another.  In  Fig.  83 


FIG.  83. 

suppose  we  have  the  diagrams  of  two  valves,  one  with  an 
eccentricity  r  =  od  and  an  angle  of  advance  d  —  aod,  and  the 
other  with  an  eccentricity  r^  —  oc  and  an  angle  of  advance 
<?,  =  aoc. 

From  the  diagram,  if  ob  is  one  dead-point,  when  the 
crank  has  moved  through  an  angle  GO  =  bog  the  first  valve 
has  moved  from  its  middle  position  a  distance  oe,  and  the 

137 


138  VAL  VE-GEARS. 

second  has  moved  in  the  same  direction  from  its  middle 
position  a  distance  of,  and  the  distance  between  the  centres 
of  the  valve  is 

fe  =  oe  —  of. 

128.  Relative  Valve-circle.  —  As  seen  from  the  first  part 
of  the  work, 

oe  =  r  sin  (GO 
ana 


and 

ef=r  sin  (GO  -f-  6)  —  rl  sin  (GO  -f-  ^) 

=  r  sin  co  cos  d  -f-  r  cos  f»  sin  d  —  rl  sin  G?  cos  ^ 

—  r4  cos  GO  sin  tf, 

=  (r  cos  ^  —  r1  cos  #,)  sin  GO  -\-  (r  sin  tf  —  rl  sin  c^)  cos  GO. 
The  value  of  */"  can   be  represented  by  a  circle,  the  co- 
ordinates of  the  centre  being 

r  sin  d  —  r.  sin  d.  r  cos  d  —  r.  cos  (f, 

and  -  , 

2  2 

and  the  other  extremity  of  the  diameter  through  o  has  co- 
ordinates r  sin  d  —  rl  sin  dl  and  r  cos  ^  —  r1  cos  ^  . 

In  the  figure,  if  we  draw  dh  parallel  to  ao  and  ch  parallel 
to  oby  the  line 

dh  —  r  cos  d  —  rl  cos  <^ 
and 

^  —  r  sin  d  —  rl  sin  ^,  , 

and  the  line  from  d  to  c  is  therefore  equal  in  length  and 
parallel  to  the  diameter  of  the  circle,  which  shows  the  length 
of  efior  every  value  of  GO,  and  this  we  will  call  the  relative 
valve-circle. 

129.  To    draw   the   Relative    Valve-circle.  —  Drawing 
through  o  the  line  ok  parallel  and  equal  to  dc,  we  have  the 


RELATIVE   MOVEMENT— POLONCEAU   GEAR.  139 

diameter  of  the  relative  valve-circle  in  its  proper  place. 
Drawing  the  circle  on  ok  as  a  diameter,  we  have  a  diagram 
showing  the  relative  motion  of  the  two  valves.  That  is, 
when  the  crank  gets  to  any  position,  as  og,  the  distance  ol  is 
the  distance  between  the  centres  of  the  two  valves. 

130.  The  Polonceau  Valve. — Referring  now  to  Fig.  77, 
when  the  valves  have  moved  so  that  the  distance  between 
their  centres  is  s,  the  valve  d  covers  the  steam-passage  and 
steam  is  cut  off.     If  now,  in  Fig.  83,  with  o  as  a  centre  we 
draw  a  circle  qv  with  s  as  a  radius,  steam  is  cut  off  when  the 
crank  reaches  ov,  and  the  port  opens  again  when  the  crank 
reaches  oq.     If  now  oq  is  later  than  the  point  of  cut-off  of  the 
main  valve,  steam  is  admitted  only  once  during  each  stroke, 
and  the  movement  is  correct.     An  examination  of  Fig  83 
shows  that  if   by  any  means   we  can   change  the  angular 
advance  or  travel  of  either  eccentric  or  valve  we  can  change 
the  position  and  size  of  the  relative  valve-circle,  and  conse- 
quently change  the  point  of  cut-off. 

131.  The  Polonceau  Gear. — Perhaps  as  simple  a  way  of 
changing  the  travel  of  the  valve  and  angular  advance  as  any 
is  by  using  a  link-motion.     Let  Fig.  84  be  a  Gooch  motion 


FIG.  84. 

arranged  with  two  radius-rods,  so  that  the  main  valve  is 
attached  to  one  and  the  cut-off  valve  to  the  other.  If  the 
radius-rods  are  at  the  same  point  on  the  link,  the  valves 
move  together  and  the  cut-off  is  that  due  to  the  mam  valve. 

132.  Valve  Diagram.— The  valve-diagram  is  that  given 
in  Fig.  85  by  the  circle  i,  and  the  cut-off  takes  place  at  oa. 

If  now  the  radius-rod  of  the  cut-off  valve  is  lowered,  say 


140 


VALVE-GEARS. 


.half-way  to  the  centre,  the  diagram  for  the  main  valve  is 
that  given  by  the  circle  i  and  for  the  cut-off  valve  that 
-given  by  2,  and  the  relative  valve-circle  is  found  by  laying 


FIG.  85. 

off  od  equal  and  parallel  to  fe.  As  for  a  Gooch  link  fe  is 
parallel  to  on,  the  centre  of  the  relative  valve-circle  is  on  on. 
Drawing  the  relative  valve-circle,  and  drawing  a  circle  gpqr 
with  s  as  a  radius,  then  og  is  Jthe  cut-off  for  the  main  valve 
u  —  c,  the  cut-off  valve  u1  =  \c. 

133.  Limits  of  Cut-off. — If  for  the  cut-off  valve  we  had 
made  «,  greater  than  $c  for  the  particular  dimensions 
selected,  the  point  d  would  have  fallen  lower  on  the  line  on, 
-and  the  circle  with  s  as  a  radius  would  have  cut  the  relative 
a  second  time  before  the  main  valve  cuts  off,  or 


RELATIVE  MOVEMENT— POLONCEAU  GEAR.  14! 

steam  would  have  been  admitted  twice  during  one  stroke. 
Evidently  the  cut-off  at  og  is  the  latest  cut-off  possible  with 
this  particular  gear. 

When  u^  =  o  for  the  cut-off  valve,  the  centre  of  the  rela- 
tive valve-circle  is  at  d  and  the  cut-off  is  at  op.  For  the  cut-off 
valve  MI  =  —  %c,  m  is  the  centre  of  the  relative  valve-circle, 
and  the  cut-off  is  at  oq.  For  u^  =  —  c  for  the  cut-off  valve- 
circle  the  centre  of  the  relative  valve-circle  is  at  n,  and  the 
cut-off  takes  place  at  or. 

As  the  radius-rod  can  be  lowered  no  further,  it  is  evident 
that  or  is  the  earliest  cut-off  attainable  with  this  gear.  The 
cut-off  is  therefore  limited  to  the  distance  between  the  two 
positions  of  the  crank  or  and  og,  and  is  therefore  of  quite 
restricted  use  except  for  engines  requiring  a  large  amount 
of  power  when  starting,  at  which  time  the  cut-off  can  take 
place  by  the  main  valve,  and  afterwards  much  less  power 
can  be  used,  as  the  engine  gets  down  to  its  steady  load. 

With  a  link-motion  actuating  both  valves  in  this  way  the 
latest  point  of  cut-off  by  the  cut-off  valve  is  at  such  a  position 
that  the  distance  from  the  point  of  cut-off  to  mid-stroke  is  the 
same  as  the  distance  from  mid-stroke  to  the  point  of  cut-off  by 
the  main  valve. 

134.  Dimensions  of  Valve. — The  area  of  the  passages  b 
and  c  both  at  the  top  and  bottom  of  the  main  valve  should 
be  as  great  or  greater  than  the  area  of  the  main  steam-port. 
The  length  of  d,  Fig.  77,  should  be  so  great  that  the  left- 
hand  edge  of  d  should  never  pass  the  left-hand  edge  of  b. 
Calling  e  the  width  of  the  port  in  b  and  rx  the  relative  valve- 
circle  diameter,  the  length  of  d>  rx  —  (s  —  e)  or  <^>  rx-\-e  —  s. 

The  distance  between  the  steam-passages  on  the  top  of 
the  main  valve  =  d-\-  2(s  —  e). 

The  outer  wall  of  the  steam-passage  b  should  be  so  wide 
that  for  the  greatest  relative  travel  the  right-hand  edge  of  d 
should  pass  over  the  end  of  the  valve  or  be  <  rx  —  s. 


142  VA  L  VE-  GEA  RS. 

QUESTIONS. 

163.  What  is  meant  by  the  relative  valve-circle,  and  how 
is  it  determined  ? 

164.  Prove  that  the  relative  movement  of  the  two  valves 
is  given  by  the  relative  valve-circle. 

165.  Describe  the  Polonceau  valve. 

1 66.  What  arrangement  was  used  for  moving  the  Polon- 
ceau valves? 

167.  How  is  the  valve-diagram   for  a  Polonceau  valve 
and  gear  drawn  ? 

1 68.  What  are  the  limits  of  cut-off  with  this  gear? 

169.  How  are  the  dimensions  of  the  main  and  cut-off 
valves  determined? 

PROBLEMS. 

64.  A   Polonceau   valve  is  driven   by  a  Gooch   motion 
having  r  —  2|  inches,  g  —  57  inches,  d  —  27°,  and  lead  equal 
to  \  inch,  and  for  u  =  6  the  cut-off  just  opens  as  the  main 
valve  closes.     Length  of  link   12  inches.     Draw  a  diagram 
showing  the  cut-off  for  u  =  6  inches,  #,  =  2,  4,  and  6  inches. 

65.  Determine  the  greatest  and  least  limit  of  the  cut-off 
with  the  data  of  Problem  64. 

66.  If  the  port  in  the  main  valve-seat  is  if  inches  and  in 
the  cut-off  valve-seat  is  if  inches,  with  the  value  of  s  as  found 
for  Problem   64  draw  the  valve  in  the  position   it  would 
occupy  if  GO  —  40°  and  the  cut-off  at  £  stroke  ;   the  exhaust- 
port  in  the  main  valve-seat  being  4  inches  and  the  bridges 
i  j-  inches,  the  iron  being  -J  inch  thick  in  the  valves. 


CHAPTER  XVII. 
BUCKEYE  GEAR. 

135.  The  Valve. — The  Buckeye  Automatic  Engine  valve 
resembles  the  valve  we  have  been  dealing  with,  but  its 
method  of  action  is  entirely  different.  Fig.  86  is  a  part  sec- 
tion through  the  valves  and  cylinder,  a  is  the  main  valve, 
having  ports  £,  through  which  steam  passes  from  the  chest 
h  to  the  passage  c  in  the  cylinder,  e  and  e  are  two  blocks 

k\\\\> 


FIG.  86. 

connected  by  rods /passing  through  an  opening  in  the  main 
valve,  and  forming  together  the  cut-off  valve.  As  steam  is 
admitted  through  //,  and  the  exhaust  takes  place  at  g,  the 
action  of  the  main  valve  is  the  same  as  of  an  ordinary  D- 
slide  taking  steam  inside  and  exhausting  outside. 

136.  The  Eccentrics  and  Connections. — In  Fig.  87  if  ab 
is  the  crank  turning  as  shown  by  the  arrow,  an  eccentric  ac 


FIG.  87. 

connected  directly,  by  an  eccentric-rod  cd,  to  a  valve-stem 
de  would  give  the  mam  valve  the  proper  motion. 

At  the  point  d,  the  eccentric-rod  is  also  connected  to  the 

143 


1 44  VAL  VE- 

lever  df,  which  is  pivoted  to  the  frame  of  the  engine  at  ft 
i  is  the  middle  of  df,  and  to  i  is  attached  the  centre  of  a 
vibrating-lever  hj,  one  end  of  which,  /,  is  connected  to  the 
cut-off  valve-stem /£,  and  the  other  end,  h,  is  connected  by 
the  eccentric-rod  hg  to  the  cut-off  eccentric  ag.  Assume 
that  ac  and  ag  are  in  their  middle  position  when  at  ac'  and 
ag' .  Call  c'ac  =  tf  and  g'ag  =  d, .  Call  ac  =  r  and  ag  =  rl  (in 
the  Buckeye  engine  these  are  equal).  Evidently,  if  both 
eccentrics  are  in  their  middle  positions  at  the  same  time, 
the  valves  would  occupy  the  positions  shown  in  Fig.  86; 
but  as  the  eccentrics  are  never  in  their  mid-positions  at 
the  same  time,  the  valves  and  ports  are  never,  in  the  engine, 
as  shown  in  the  figure. 

137.  Movement  of  the  Valves. — When  the  crank  ab  has 
moved  in  the  direction  of  the  arrow  through  the  angle 
co,  if  the  eccentrics  are  in  their  proper  position  the  angle 
cac  =  d  -\-  GO  and  g'ag  =  ^  —  GO.  The  main  valve  has  moved 
to  the  left  a  distance  r  sin  (GO-}-  d).  The  point  d  has  moved 
to  the  left  the  same  distance,  and  the  point  i  has  moved  to 

the  left  —  sin  (GO  -f-  6).  If  the  eccentric  ag  is  still  in  its  mid- 
position  the  lever  hj  turns  about  h  as  a  centre,  and  j  has 
moved  to  the  left  a  distance  =  r  sin  (co-\-  #),  and  the  cut-off 
valve  has  moved  to  the  left  a  distance  r  sin  (GO  +  d)  due  to 
the  movement  of  the  main  eccentric  alone.  To  bring  the 
cut-off  eccentric  to  its  proper  position  g  moves  to  the  right 
a  distance  r,  sin  (GO  —  tfj  and  /moves  to  the  left  a  distance  rt 
sin  (GO  —  #,),  and  the  total  movement  of  the  cut-off  valve  is  r 
sin  (GO-}-  6)  -J-  rl  sin  (GO  —  <?,),  and  the  relative  movement  of 
the  two  valves  is 

x  =  r  sin  (GO  -|-  d)  -f-  r1  sin  (GO  —  tfj  —  r  sin  (GO  -f-  $) 
=  rl  sin  (GO  —  £,). 

But  this  is  the  equation  to  the  valve-circle  for  the  cut-off 
valve ;  that  is,  the  arrangement  is  such  that  the  cut-off  valve 
moves  on  the  main  valve  in  the  same  way  as  though  the 


BUCKEYE   GEAR. 


main  valve  did  not  move  at  all,  and  the  cut-off  eccentric  was 
connected  through  an  ordinary  reverse  lever  to  the  cut-off 
valve. 

138.  Cut-off  Valve-diagram. — In  Fig.  88  we  have  drawn 
the  valve  diagram  for  the  cut-off  valve 

#j  being  negative.     From   Fig.   86  we 

see  that  the  cut-off  valve  must  move  to 

the  left  a  distance  s  to  close  the  port  in 

the  steam-valve.     With  this  distance  s 

'as  a  radius  draw  the  circle  cde.     The 

cut-off  takes  place  when  the  crank  is  at 

oc  and  the  port  is  again  open  when  the 

crank  reaches  oe  and  remains  open  on  the  same  end  until  the 

crank  again  reaches  oc. 

139.  Changing  Cut-off.— If  the  angle  of  advance  of  the 
cut-off  valve  is  made  variable,  the  point  of  cut-off  can  be 
changed  to  any  extent  as  long  as  oe  does  not  come  before 
the  cut-off  of  the   main  valve,   and  it  is  by  changing  the 
angular  advance  of  the  cut-off  valve  that  the  cut-off  is  made 
variable  in  the  Buckeye  engine.     In  Fig.  89  let  goa  —  d,  the 


FIG. 


FIG. 


angle  of  advance  of  the  main  valve.  Draw  the  main  valve 
circle  ;  cbe  is  the  steam-lap  circle.  Steam  is  admitted  when- 
tbfl  crank  is  at  0/and  cut-off  at  ok.  The  circle  jm/i  is  described 
with  s  as  a  radius.  The  earliest  possible  cut-off  would  be 
when  the  valve-circle  for  the  cut-off  valve  passes  through  o> 


146 


VAL  VE-GEARS. 


and  m.  This  will  be  found  to  be  at  or  before  the  beginning 
of  the  stroke,  and  of  course  no  such  movement  of  the  eccen- 
tric is  necessary.  If  the  angle  of  advance  ^  is  zero,  the 
cut-off  takes  place  at  on,  and  the  port  is  again  open  to  steam 
at  of  after  the  main  valve  has  cut  off  at  ok.  As  dl  increases 
negatively  the  point  of  cut-off  is  later  and  later,  and  if 
necessary  could  be  made  as  late  as  with  the  main  valve  by 
allowing  the  cut-off  valve  to  change  its  angular  advance 
sufficiently. 

140.  The  Governor. — The   arrangement  by  which   the 
change  in  angular  advance  is  effected  is  shown  in  Fig.  90. 


FIG.  90. 

A.  governor-wheel  is  keyed  to  the  shaft ;  a  and  a  are  two 
arms  pivoted  to  the  arms  of  the  wheel  at  gg ;  bb  are  rods 


BUCKEYE   GEAR.  147 

joining  these  arms  to  the  lugs  cc,  which  are  attached  directly 
to  the  cut-off  eccentric.  This  eccentric  is  loose  on  the 
shaft  while  the  eccentric  for  the  main  valve  is  secured  at  its 
proper  angle  of  advance;  dd  are  weights  which,  as  the 
speed  increases,  fly  out  and  turn  the  cut-off  eccentric,  thus 
changing  the  angle  of  advance;  e  and  /are  springs  which 
act  against  the  centrifugal  force  tending  to  throw  the 
weights  outward. 

QUESTIONS. 

170.  Make  a  sketch  of  the  Buckeye  valve,  and  show  the 
relative  position  of  crank  and  eccentrics  which  would  drive 
the  valve  as  a  Polonceau  is  driven. 

171.  Make  a  sketch  of  the  eccentric  and  connections  when 
the  engine  is  intended  to  run  in  the  direction  opposite  to 
that  shown  in  Fig.  87. 

172.  Deduce  the  equation  to  the  relative  moments  of  the 
two  valves. 

173.  Draw  the  cut-off   valve-diagram,  and   explain  how 
the  point  of  cut-off  is  changed. 

174.  Sketch  and  explain  the  governor. 

PROBLEMS. 

67.  In  a  Buckeye  gear  if  the  lap  is  i^  inches,  the  port- 
opening  is  if  inches,  and  the  maximum  cut-off   is  to  take 
place  at  \  stroke,  what  must  be  the  travel  of  the  main  valve  ? 
If  the  cut-off  eccentric  has  the  same  eccentricity,  what  must 
be  the  angular  movement  of  the  cut-off  eccentric  to  vary  the 
cut-off  from  \  to  f  the  stroke,  and  what  must  be  the  actual 
value  of  #  for  cutting  off  at  f  stroke,  s  being  I  inch  ? 

68.  If  the  value  of  s  is  changed,  does  it  make  any  differ- 
ence in  the  angular  movement  of  the  eccentric?     Can  the 
value  of  s  be  made  too  large  or  too  small  ?    How  then  would 
you  choose  the  proper  value  ?    What  should  it  be  in  the  last 
problem  ? 

69.  What  must  be  the  width  of  the  cut-off  blocks  in  the 
last  problem  ? 


CHAPTER  XVIII. 


MEYER   VALVE   AND   GUINOTTE   GEAR. 

141.  The  Meyer  Valve. — We  have  seen  that  by  chang- 
ing the  position  of  the  cut-off  eccentric  the  point  of  cut-off 
can  be  changed.  There  is  another  device  by  which  this  can 
be  done,  which  is  shown  in  the  Meyer  valve,  represented 
in  Fig.  91.  a  is  the  main  valve,  having  passages  through  it 
like  the  Polonceau  valve,  b  and  c  are  the  cut-off  blocks, 


FIG.  91. 

which  together  form  the  cut-off  valve.  The  valves  are 
moved  by  independent  eccentrics  connected  directly. 

The  valve-diagram  for  this  valve  is  the  same  as  for  the 
Polonceau.  b  and  c  are  connected  by  a  right-  and  left- 
handed  screw  by  means  of  which  the  distance  between  the 
blocks  is  regulated. 

142.  Changing  the  Distance  between  the  Blocks.— To 
determine  what  effect  the  changing  of  the  distance  between 

148 


MEYER    VALVE  AND   GUINOTTE   GEAR. 


149 


the  blocks  has  on  the  point  of  cut-off,  let  Fig.  92  be  the  valve- 
diagram,  supposing  the  main-valve  eccentric  to  be  set  with 
an  angle  of  advance  abe,  and  the  cut-off  eccentric  with  an 
angle  of  advance  abd.  Then,  as  already  shown,  for  the 
Polonceau  gear  the  relative,  valve-circle  is  that  shown  at 


FIG.  92. 

cfghb.  The  cut-off  takes  place  when  the  distance  be,  Fig.  92, 
is  equal  to  s  of  Fig.  91. 

Suppose  now  in  Fig.  91  that  the  blocks  b  and  c  are  moved 
closer  together.  The  distance  s  in  Fig.  91  is  increased,  and 
in  Fig.  92  if  bf—  the  new  value  of  s,  the  cut-off  now  takes 
place  at  bj.  As  in  a  gear  of  this  kind  it  is  convenient  to 
measure  the  distance  between  the  blocks,  let  it  be  2y.  Let 
the  length  of  b  or  c  be  d,  and  the  distance  between  the  out- 
side edges  of  the  ports  be  L.  Then  2s  —  L  —  2d  —  2y. 

If  it  is  required  to  know  at  what  point  the  blocks  should 

be  for  any  point  of  cut-off  :  2y  =  L  —  2d  —  2s,  or  y  = d 

-  s.     Drawing  a  circle  ijh',  Fig.  92,  with  —  —  d  as  a  radius, 

at  any  point  of  cut-off,  as  be,  we  have  be  = d  =  y  -f-  s, 

and  be  =^  s,  ce  =  y.    As  the  cut-off  becomes  earlier  and  earlier 


ISO 


VAL  VE-GEARS. 


at  bl,  we  have  s  —  o  and  y  =  bl\  and  as  the  cut-off  is  still 
earlier  as  bm,  s  is  negative  and  —  —  bn  and  y  =  ;#;/.  So 
that  if  the  value  of  y  is  changed,  almost  any  point  of  cut-off 
desired  can  be  obtained. 

It  is  to  be  remembered  that  the  pointy  or  //,  in  which  the 
circle  with  s  as  a  radius  cuts  the  relative  valve-circle  a 
second  time,  must  not  occur  before  the  main  valve  cuts  off, 
otherwise  steam  is  admitted  twice  during  the  same  stroke. 

143.  Designing  a  Meyer  Valve. — In  designing  a  gear  of 
this  kind,  the  main  valve  is  designed  for  the  latest  cut-off 
desired,  and  the  range  over  which  the  cut-off  valve  is  in- 
tended to  act  is  determined.  Suppose  in  Fig.  93  cab  to  be 


FIG.  93. 

the  angle  of  advance  of  the  main  valve,  ab  its  half-travel,  and 
ae  the  lap.  Steam  is  admitted  at  af,  and  cut  off  by  the  main 
valve  takes  place  at  ag.  Now  suppose  it  is  required  to 
design  a  Meyer  gear  to  act  between  ah  and  ag.  As  the  cut- 
off is  to  range  entirely  up  to  that  of  the  main  valve,  and  as 
we  do  not  want  any  larger  eccentric  than  is  necessary,  we 
will  assume  that  the  relative  valve-circle  centre  falls  on  ag. 

We  will  assume  further  that  the  engine  is  intended  to  run 
in  either  direction  equally  well.  That  this  may  be  so,  the 
centre  of  the  valve-circle  for  the  cut-off  valve  should  be  on 
at.  Through  b  draw  bi  parallel  to  ag.  Then  at  is  the  eccen- 
tricity of  the  cut-off  valve-eccentric,  and  bi  is  the  equivalent 
of  the  diameter  of  the  relative  valve-circle.  We  have  seen 
that  when  s  is  greatest  the  cut-off  is  latest,  or  s  must  be 


t 
MEYER    VALVE  AND   GUINOTTE   GEAR.  151 

greatest  at  ag.     The  blocks  can  here  be  close  together,  or 
y  =  o,  and  we  have  -  —  d  =  s  =  bi.    When  s  is  least  the  cut- 

off is  earliest  ;  and  we  have  from  the  diagram  the  value  of 

L 

s  =  —  ak  =  —  —  d  —  y, 

or 

y  —  4-  ak  -\-  ^-  —  d\  =  ak  +  bi  =  ak  +  ah  =  kh. 

From  Fig.  91  ,—  =  y  -\-  d  -{-  s,  or  d  -\-  s  =  —    -  y. 

2  2 

144.  Length  of  Cut-off  Blocks.—  The  relative  valve-mo- 
tion should  never  be  so  great  that  the  left-hand  edge  of  c 
uncovers  the  left-hand  edge  of  the  passage  in  the  right-hand 
end  of  the  main  valve.  Calling  the  port  at  the  top  of  this 
passage  e,  we  have 

d-\-  s  —  rx  >  *,    or    -  —  y  —  rx  >  e, 
or 

-> 

Now  the  greatest  value  of  y  is  kh,  so  that  —  should  be  greater 
than  e  -j-  kh  -f-  ah.     This  determines  the  value  of  —  ,  and 


if  the  blocks  are  to  come  together  for  greatest  cut-off.  The 
value  of  y  depends  on  the  desired  point  of  cut-off,  and  is 
determined  as  already  shown. 

145.  Cut-off  with  Inside  Edges.  —  We  have  made  the 
valve§  cut  off  with  the  outside  edges.  By  setting  the  eccen- 
trics properly,  the  cut-off  could  have  been  effected  by  the 
inside  edges;  but  any  such  arrangement  is  of  no  advantage, 
as  the  main  valve  and  chest  are  increased  in  length,  and  no 
better  distribution  of  steam  is  effected. 


152 


VALVE-GEARS. 


146.  Guinotte  Gear. 


There  are  numerous  other  ways  in 

J 

which  the  variation  of  cut- 
off can  be  effected  with  a 
Polonceau  valve  or  a  modi- 
fication of  it,  but  we  will 
only  take  up  one  more. 
Referring  to  Fig.  85,  as  a 
Gooch  link  is  there  used, 
the  centres  of  the  valve- 
circles  move  in  a  line  at 
right  angles  to  os,  and  the 
centres  of  the  relative  valve- 
circles  fall  on  the  line  om. 
Now  if  by  any  means  the 
centre  of  the  cut-off  valve- 
circle  can  be  made  to  move 
at  any  other  angle  than  a 
right  angle  to  os,  the  centre 
<£  of  the  relative  valve-circle 
will  no  longer  always  be  on 
om,  and  a  greater  variation 
of  cut-off  can  be  obtained. 
There  are  two  methods  by 
which  this  can  be  done,  one 
of  which  is  given  here. 
Fig.  94  represents  a  Gooch 
motion  similar  to  Fig.  46, 
except  that  the  eccentrics 
are  set  with  unequal  angles 
of  advance,  and  the  unequal 
eccentrics,  lengths  of  eccen- 
tric-rods, and  unequal  parts 
into  which  the  link 'is  di- 
vided at  the  point  of  sus- 
pension, replace  the  cor- 
responding equal  ones  in 
Fig.  46. 


f 

MEYER    VALVE  AND   GUINOTTE   GEAR.  153 

147.  Movement  of  the  Valve.  —  We  have,  as  in  that  case 
(calling  everything-  pertaining  to  the  upper  end  of  the  link 
sub  i  (  ,  )  and  to  the  lower  end  sub  2  (^  )  g  moves  to  the  right 
because  of  the  movement  of  the  upper  eccentric, 


and  because  of  the  movement  of  the  lower  eccentric, 


and  the  total  movement  of  g  is  therefore  the  sum  of  these 
two,  or 

*  =          r>  sin  <M  +  6>  +  *>  +          r<  sm  <*•  -  *  +  *>• 


This  can  be  expanded  as  in  the  case  of  the  Gooch  motion, 
and  becomes 


7LZTr«  (sin  *•  +  7"  cos  *•)  f 

6i  ~r  62      \  s*  i  ) 


cos 


—  ——  —  r2  fcos  d  2  —  —  sin  (^,j  >•  sin  co; 


and  we  have  for  the  valve-diagram 

x  —  A  cos  GO  +  B  sin  G?, 

in  which  ^4  and  B  are  given  from  the  preceding  equation. 
The  first  power  of  u  appears  in  both  of  these  values,  and 
therefore  the  centre  of  the  valve-circles  moves  in  a  straight 
line  inclined  to  os. 


154  VALVE-GEARS. 

148.  To  draw  the  Valve-diagram.  —  To  draw  the  line  of 
centres  a  convenient  method  is  as  follows:  First  suppose 
u  —  c  ;  then 


and 


which  gives  the  centre  of  the  valve-circle  for  an  ordinary 
Gooch  motion  having  data  r1 ,  dl ,  cl ,  and  gl  when  u  =  cl . 
Now  let  u  =  —  £a ,  and  we  have 


and 


which  gives  the  centre  of  the  valve-circle  for  an  ordinary 
Gooch  motion,  with  data  r^ ,  d^ ,  £2  and^  when  u  =  —  c^ .  The 
line  can  then  be  drawn  through  these  points  and  the  points 
of  cut-off  for  different  values  of  u  for  the  cut-off  radius-rod 
found,  the  main  valve-eccentric  having  data  r  and  d. 

Fig.  95  shows  the  construction  of  the  diagram  for  the 

main  valve  and  for  the  cut-off  valve  when  u  =  ct ,  -1 ,  o,  and 

—  £2.  As  this  form  of  gear  is  only  of  use  for  engines  run- 
ning in  one  direction,  it  is  not  at  all  likely  to  be  used  to  any 
extent. 

QUESTIONS. 

175.  Sketch  a  Meyer  valve,  and  explain  the  method  of 
drawing  the  valve-diagram. 

176.  Explain  from  the  diagram  the  effect  of  changing  the 
distance  between  the  cut-off  blocks. 


A,  =  r,  (sin  d,  +  ^  cos  £,, 

\  o  a 


^  —  —  r^  [cos  £t  —  —  sin  <?2j, 


MEYER    VALVE  AND   GUINOTTE   GEAR. 


155 


177.  How  determine  from  the  diagram  the  distance  be- 
tween the  blocks  for  any  given  point  of  cut-off? 

178.  If  one  block  is  moved  farther  from  the  centre  than 
the  other,  will  the  cut-off  on  the  two  ends  be  the  same? 
Why  ?     Could  such  a  device  be  used  to  make  the  engine 
cut-off  at  exactly  the  same  distance  on  each  end  for  different 
points  of  cut-off? 

179.  Explain  the  method  of  designing  a  Meyer  valve. 


1 80.  How  determine  the  width  of  the  blocks  and  dimen- 
sions of  the  top  of  the  main  valve? 

181.  What  would  be  the  advantage  in  cutting  off  with 
the  inside  edges  of  the  blocks? 

182.  Make  a  sketch  of  the  Guinotte  gear,  and  explain 
how  it  is  used. 

183.  Deduce  the  equation  to  the  movement  of  the  cut-off 
valve. 

184.  Draw  the  valve-diagram  for  different  points  of  cut- 
off. 


156  VALVE-GEARS. 


PROBLEMS. 

70.  Given  for  a  Meyer  valve,  the  main  valve  cuts  off  at 
^f  stroke,  has  an  eccentricity  of   if  inches,  and  the  angle  of 
lead  is  8°.     If  the  port  e  —  £  jnch,  and  rx  =  i$  inches,  find 
the  length  of  the  cut-off  blocks,  L,  6,  and  r  for  the  cut-off 
eccentric,  and  y  for  cutting  off  at  \  and  f  stroke,  the  maxi- 
mum cut-off  with  the  blocks  being  at  f  stroke. 

71.  Given   r  —  if   inches,   <5=  15°,  r,  =  i%   inches,   and 
d1  =  85°.     Draw  the  relative  valve-circle,  and  determine  the 
point  of  cut-off  if  s  =  i|  inches.     If  the  main  valve  is  moved 
by  a  Stephenson's  link,  in  which  g  =  60  inches  and  2c  =  14 
inches,  where  is  the  cut-off  for  u  =  3  and  5  inches. 

72.  Given  the  stroke  30  inches,  connecting-rod  60  inches, 
r  for    both   valves    2j   inches,    6  =  30°,    tf,  =  90°,   L  =  17 
inches,  e  =  2  inches,  d—  5f  inches.     It  is  required  that  the 
cut-off  should  take  place  exactly  at  \  stroke  and  at  f  stroke 
in  each  end,  what  must  be  the  value  of  y  for  each  block  for 
•each  point  of  cut-off. 

73.  Given  for  a  Guinotte  gear  r  =  2f  inches,  d  =  22%°, 
rl  =  2f    inches,    #,  =  30°,  gl  =  60  inches,    cl  ==  4f    inches, 
r^  =  i j-  inches,  #3  =  45°,  g^  =  50  inches,  and  c^  =  5j  inches. 
Draw  the  valve-diagram,  and  show  the  point  of  cut-off  for 
u  =  cl ,  o,  and  —  ct . 


CHAPTER  XIX. 

BILGRAM,    REULEAUX,  AND   ELLIPTICAL   DIAGRAMS. 

149.  Bilgram  Diagram. — Many  other  diagrams  are  used 
to  represent  the  movement  of  the  valve,  none  of  which  is  as 
convenient  as,  and  none  more  accurate  than  the  Zeuner  dia- 
gram. The  simplest  of  these  is  the  Bilgram  diagram,  which 
is  represented  in  Fig.  96. 


FIG.  96. 

Draw  oa  and  ob  at  right  angles  to  each  other,  and  with  r 
as  a  radius  draw  the  circle  chd. 

Lay  off  the  angle  aod  equal  to  #,  and  with  d  as  a  centre 
and  /,  the  lap,  as  a  radius,  draw  the  lap-circle  ejg.  When  the 
crank  has  reached  the  line  oh,  the  distance  the  valve  has 
travelled  from  its  middle  position  is  given  by  a  perpendicu- 
lar from  d  on  oh,  or  the  movement  of  the  valve  x  =  di,  and 
the  opening  of  the  port  is  ji.  For  di  =  od  sin  doi  =  r  sin 


The  port-opening  takes  place  when  the  crank  position  is 

157 


i58 


VALVE-GEARS. 


tangent  to  the  lap-circle  as  at  of,  and  cut-off  takes  place 
when  the  crank  position  is  again  tangent  to  the  lap-circle,  as 
at  eg,  because  in  these  positions  the  perpendicular  from  d  to 
the  crank  line  is  equal  to  the  lap.  Drawing  dk  perpendicu- 
lar to  oa  gives  dk  equal  to  the  lap  plus  the  lead,  as  it  is  the 
distance  the  valve  has  moved  from  its  central  position  when 
the  crank  is  on  a  dead-point,  and  ek  is  the  lead. 

150.  Problems. — The  problems  connected  with  the  sim- 
ple valve  can  be  solved  as  easily  with  this  diagram  as  with 
the  Zeuner  diagram,  and  the  solution  of  each  is  given  below. 

PROBLEM  i.  Given  r,  d,  the  point  of  cut-off,  and  the  point 
of  closing  of  the  exhaust,  to  find  the  lap,  exhaust-lap,  lead, 
and  exhaust-lead,  and  the  greatest  possible  opening  of  the 
port.  In  Fig.  97  draw  oa  and  ob  at  right  angles,  and  draw 


FIG.  97. 

the  circle  abc,  with  r  as  a  radius.  Lay  off  doa  =  d,  o%  for  the 
point  of  cut-off,  and  ol  for  the  point  of  closing  of  the  ex- 
haust. Draw  dg  perpendicular  to  og,  and  dl  perpendicular 
to  ol\  then  dg  is  the  lap  and  dl  the  exhaust-lap.  Draw  the 
lap  and  exhaust-lap  circles ;  then  ek  is  the  lead,  mk  is  the 
exhaust-lead,  no  is  the  greatest  opening  of  the  port  to  steam, 
and  op  to  exhaust. 

PROBLEM  2.  Given  the  lap,  point  of  cut-off,  and  lead,  to 
determine  the  eccentricity  and  angular  advance. 

In  Fig.  98  lay  off  ogior  the  point  of  cut-off,  and  at  any 
point  k  draw  kd  at  right  angles  to  ao,  and  equal  to  the  lap 
plus  the  lead.  Draw  the  lap-circle  with  d  as  a  centre,  and 
draw  a  line  o'g*  parallel  to  og  and  tangent  to  the  lap-circle ; 


BILGRAM,   REULEAUX,    AND  ELLIPTICAL  DIAGRAMS.    I  $9 

then,  joining  d  and  or,  the  angle  do' a  is  the  angular  advance, 
and  do'  is  the  eccentricity. 

PROBLEM  3.  Given  the  cut-off,  angle  of  lead,  width  of 


FIG.  98. 

port,  and  the  overtravel,  to  determine  the  eccentricity,  lap, 
lead,  and  angular  advance. 

Lay  off  og1  and  of  in  Fig.  99  to  represent  the  crank  at 
cut-off  and  at  the  opening  of  the  port.     The  centre  of  the 


FIG.  99. 

lap-circle  will  lie  on  the  bisector  of  the  angle  between  eg  and 
*?/ produced,  or  on  od.  Lay  off  on  equal  to  the  width  of  the 
port  plus  the  overtravel,  and  draw  ng' ,  making  an  angle  of 
90°  with  og' .  Make  nn'  equal  to  ngr,  draw  n'g',  and  ng  par- 
allel to  rig'.  Draw  gd  parallel  to  ng*\  and  d,  the  intersection 
with  od,  is  the  centre  of  the  lap-circle.  Draw  dk  parallel  to 
ob ;  then  od  is  the  eccentricity,  dg  is  the  lap,  ek  is  the  lead,  and 
dok  is  the  angular  advance. 

PROBLEM  4.    Given  the  point  of  cut-off,  lead,  and  port- 


VALVE-GEARS. 


opening-,  to  determine  the  angular  advance,  lap,  and  eccen. 
tricity. 

In  Fig.   100  lay  off  og  to  represent  the  point  of  cut-offv 
Draw  ee'  parallel  to  oa,  so  that  ek  equals  the  lead. 


FIG.  100. 

Draw  an  arc  nri  with  the  port-opening  as  a  radius.  Find 
by  trial  such  a  point  d  that  a  circle  drawn  with  */as  a  centre 
will  be  just  tangent  to  ee' ,  nn' ,  and  og.  doa  is  then  the  angu- 
lar advance,  dn  is  the  lap,  and  od  is  the  eccentricity. 

151.  Reuleaux's  Diagram. — The  diagram  constructed  by 
Professor  Reuleaux  is  equally  convenient  for  the  solution  of 

b 


FIG.  101. 
most  of  the  problems  connected  with  simple  valves. 


represented  in  Fig.  101. 


It  is 
Draw  oa  and  ob  at  right  angles  to 


BILGRAM,   REULEAUX,   AND  ELLIPTICAL   DIAGRAMS.    l6l 

each  other,  and  make  aof  and  boe  equal  to  d.  Make  od  equal 
to  the  lap,  and  draw  dg  parallel  to  of.  Starting-  from  the 
dead-point  oa  when  the  crank  has  moved  an  angle  GO  to  the 
position  ok,  the  valve  has  moved  a  distance  hi  from  its  cen- 
tral position,  hi  being  parallel  to  oe.  For 

hi  =  oh  sin  hoi  —  r  sin  ($  -f-  GO). 

The  port-opening  is  hj.  The  lead  is  found  by  drawing  ak 
parallel  to  oe,  as  this  is  the  port-opening  when  GO  =  o.  de  is 
the  maximum  port-opening.  Cut-off  takes  place  at  ol,  and 
steam  is  admitted  at  og. 

152.  Problems. — The  following  is  the  method  of  solving 
some  of  the  problems  already  given  for  a  simple  valve  by 
Reuleaux's  diagram  : 

PROBLEM  i.  Given  r,  6,  the  point  of  cut-off,  to  find  the 
lap,  lead,  and  greatest  port-opening.  Draw  oa  and  ob  at 
right  angles  in  Fig.  102,  and  lay  off  boe  =  aof  '=•  d.  Draw 


FIG.  102. 

the  circle  aeb,  with  o  as  a  centre  and  r  as  a  radius.  Lay  off 
ol  for  the  point  of  cut-off.  Draw  Ig  parallel  to  of,  and  ak 
parallel  to  eo.  Then  od  =  lap,  ak  =  lead,  and  ed  is  the  great- 
est port-opening. 

PROBLEM  2.  Given  lap,  point  of  cut-off,  and  lead,  to  de- 
termine the  eccentricity  and  angular  advance. 


162 


VALVE-GEARS. 


In   Fig.    103  draw  ol  to  represent  the  point  of  cut-off. 

am       lead 

Draw  the  circle  abc   with  any  radius.      Make  -  -  =  -, — . 

mo        lap 

Draw  the  lines  Img  through  /  and  m,  and  ak  and  oe  at  right 
angles  to  gl.    Then  boe  is  the  angular  advance,  and  the  figure 


FIG.  103. 

is  drawn  to  such  a  scale  that  ak  represents  the  lead,  or  the 

eo  X  lead 
eccentricity  is v ,  or  a  second  diagram  can  be  drawn 

with  dimensions  — 7-  larger  than  the  one  in  the  figure,  from 

Cl  K 

which  the  eccentricity  can  be  directly  measured. 

PROBLEM  3.  Given  the  cut-off,  angle  of  lead   width  of 


FIG.  104. 

port,  and  overtravel,  to  determine  the  eccentricity,  lap,  lead, 
and  angular  advance. 


BILGRAM,   REULEAUX,   AND  ELLIPTICAL  DIAGRAMS.    163 

In  Fig.  104  lay  off  olf  for  the  cut-off  and  aog'  for  the  angle 
of  lead.  Bisect  the  angle  log'  by  oe' ;  then  b'oe'  is  the  angular 
advance.  With  any  radius  as  oe' ,  draw  g'e 'I'  and  draw  g'l'. 
Then  as  d'e'  is  to  the  real  port-opening,  so  is  oe'  to  the  real 
half-travel.  Find  the  half-travel  oe,  draw  the  circle  gael,  and 
the  line  gL  Then  ak  is  the  lead,  od  the  lap,  and  oe  the 
eccentricity. 

153.  Elliptical  Diagrams. — It  is  sometimes  convenient 
to  represent  the  movement  of  the  valve  as  compared  with 
that  of  the  piston  on  rectangular  axes,  as  shown  in  Fig.  105. 


FIG.  105. 

When  the  crank  reaches  oa  the  piston  has  moved  a  distance 
eb.  -  ob  —  R  cos  oj.  If  on  ab  we  lay  off  bd  =  r  sin  (GO  -f-  tf), 
the  locus  of  d  is  a  curve  whose  coordinates  are 

y  —  r  sin  (G?  -f-  tf)     and     x  =  R  cos  GO, 

which  is  an  ellipse. 

If  the  angularity  of  the  connecting-rod  and  of  the  eccen- 
tric-rod are  taken  into  account,  as  they  should  be  to  use  the 
diagram  satisfactorily,  the  curve  is  only  approximately  an 
ellipse. 

Draw  hf  parallel  to  oe  so  that  be  is  the  lap.  Then  cd  is 
the  opening  of  the  port.  At  h,  where  hf  cuts  the  ellipse, 
draw  hi  at  right  angles  to  oe ;  then  oi  is  the  point  of  cut-off. 


164 


VAL  VE-GEARS. 


Similarly,  from  the  point  n  determine  om  for  the  point  of 
admission.  The  lead  is  the  distance  fg. 

154.  Velocity  of  the  Valve. — With  the  Zeuner  diagram 
the  velocity  of  the  valve  can  be  readily  determined  by  a 
similar  circular  diagram. 

In  Fig.  106,  if  the  circle  on  ob  is  the  valve-diagram,  draw 


FIG.  106. 

oc  at  right  angles  to  ob,  and  draw  an  equal  circle  on  oc  as  a 
diameter.  As  od  represents  the  movement  of  the  valve,  oe 
represents  the  velocity  of  the  valve.  For 


x  =  r  sin  (GO 


and 


^ 
doo 


=  r  cos 


=  oe, 


BILGRAM,   REULEAUX,   AND  ELLIPTICAL  DIAGRAMS.    165 

from  the  figure.  To  find  the  velocity  in  inches  per  second, 
let  n  be  the  number  of  revolutions  of  the  shaft  per  minute. 
Then 

n  X  27T       nn 

doo  =  —  -2  --  =  —  , 

60  30' 

and 


x  =  ~p  COS 
or  multiply  oe  in  the  figure  by  —  . 

QUESTIONS. 

185.  Explain  the  Bilgram  diagram. 

1  86.  Explain  the  Reuleaux  diagram. 

187.  Show  how  to  draw  a  valve  ellipse,  and  explain  it 
fully. 

1  88.  Show  how  to  determine  the  velocity  of  the  valve, 
and  how  to  calculate  the  velocity  in  feet  per  second. 

PROBLEMS. 

Problems  18,  19,  20,  21  can  be  solved  by  the  methods  of 
this  chapter  either  by  the  Bilgram  or  Reuleaux  diagram. 

74.  In  Problem  70,  how  fast  is  the  cut-off  valve  moving 
at  f  cut-off  if  n  =  60? 

75.  Given  r  —  3J-  inches,  d  =  30°,  lap  ==  i  J  inches.    What 
is  the  velocity  of  the  valve  in  feet  per  second  at  GO  =  o°,  30°, 
45°,  if  the  engine  makes  120  turns  per  minute. 


CHAPTER   XX. 

CORLISS  VALVE-GEAR. 

155.  Hamilton-Corliss  Engine. — The  Corliss  engine  has 
four  valves,  two  for  steam  on  the  upper  side  of  the  cylinder 
and  two  for  exhaust  on  the  bottom.  Fig.  107  is  a  line  dia- 
gram of  the  Hamilton-Corliss  engine  in  the  Mechanical 
Engineering  Laboratory  of  the  University  of  Pennsylvania, 
and  Fig.  108  represents  a  part  section  and  part  outside  view 
of  the  cylinder. 

In  Fig.  107  O  is  the  centre  of  the  shaft,  the  crank  being 
on  one  dead-point  at  a.  The  eccentric  is  at  Ob,  the  engine 
turning  in  the  direction  of  the  arrow.  The  eccentric-rod 
be  takes  hold  of  a  pin  c  on  a  lever  de,  which  is  pivoted  at  d 


on  the  frame  of  the  engine.  From  e  the  hook-rod  ef  takes 
hold  of  a  pin  /  on  the  wrist-plate  (g,  Fig.  108)  pivoted  at  o. 
This  wrist-plate  carries  four  studs,  h,  i,  /,  and  k,  each  of 
which  drives  one  of  the  four  valves :  ^through  im  and  jl  the 
steam-rods,  and  hn  and  kp  the  exhaust-rods.  The  valves 
are  driven  by  spindles  q,  r,  s,  and  /,  which  are  connected  to 
the  steam  and  exhaust  rods  by  the  arms  ql,  rm,  ns,  and  pt. 

The  connection  between  ns  and  pt  and  the  exhaust-valves 
is  a  permanent  one,  while  the  steam-valves  are  connected  in 
such  a  way  that  they  can  readily  be  disengaged  from  the 
driving  mechanism.  The  upper  right-hand  steam-valve  con- 
nection is  shown  in  Fig.  108.  The  arm  rm  is  carried  on  a 

166 


CORLISS    VALVE-GEAR. 


FIG.  IO&, 


1 68  VALVE-GEARS. 

loose  collar,  which  also  carries  the  arm  v.  To  this  arm  is 
attached  the  hook  wx  at  the  point  y.  This  hook  can  turn 
around  y,  and  the  arm  x  in  the  figure  is  always  kept  as  far  as 
possible  to  the  right  by  the  spring  represented  at  z.  The 
valve-spindle  r  carries  an  arm  A,  which  has  at  B  a  pin  over 
which  a  recess  in  the  hook  w  catches,  and  thus  moves  the 
arm  A  and  the  valve  connected  to  r.  C  is  a  second  loose 
collar  which  is  connected  at  D  to  the  reach-rod  DE,  which 
is  moved  by  the  governor.  The  part  C  carries  a  cam-piece 
Fj  which  as  the  arm  v  is  raised  strikes  the  inner  side  of  the 
hook-piece  x  and  causes  the  hook  w  to  disengage  the  pin  B 
and  allows  the  valve  to  be  closed  by  means  of  the  rod  Hy 
the  lower  end  of  which  is  attached  to  a  dash-pot. 

The  governor  causes  the  collar  C  and  the  cam  F  to  move, 
thus  varying  the  point  at  which  the  hook  disengages,  and 
thus  varying  the  cut-off.  The  cam  G  is  to  insure  the  hook 
w  disengaging,  if  the  arm  v  travels  too  far  downwards. 

156.  Movement  of  the  Valve. — The  movement  of  the 
end  b  of  the  eccentric,  referring  to  Fig.  107,  is  r  sin  (GO-\-  #), 
as  for  a  simple  valve.  The  movement  of  c  is  practically  the 

de 
same,  and  of  e  is  -j-r  sin  (GO -\- $).     The   movement  of  the 

point  f  is  the  same,  and  the  angular  movement  of  of  is 

T        /z/7 

—f%-j~r  si*1  (°°  H~  ^)-      The   distance   the   point  j  moves  is 

—-  -  -r  r  sin  (GO  +  <?),  and  the  movement  of  /  is  the  same  as 
of  dc 

long  as  oj  is  to  the  left  of  the  line  og  in  the  figure.  But  the 
movement  of  the  valve  is  less  than  the  distance  moved  by 

**O  |^1 

/,  or  is  — j-  times  as  much,  and  consequently  the  movement 
of  the  valve  or 

rad.  valve    oj  de 

x  = -, -r'-,-r  sin  (GO  4-  £). 

ql  of  dc 

The  same  formula  will  hold  good  for  the  exhaust-valves  as 


CORLISS    VALVE-GEAR. 


169 


long  as  any  two  of  the  three  parts  ok,  kp,  and  //  do  not 
nearly  form  one  straight  line. 

The  steam-valves  should  therefore  open  on  that  portion 
of  their  motion  where  j  is  moving  away  from  oq  and  ql  is 
approaching  it,  and  the  exhaust-valves  should  open  while  k 
is  moving  away  from  ot  and//  is  approaching  it. 

As  in  the  case  of  a  plain  slide-valve,  the  steam-port  opens 
when  the  valve  has  moved  from  its  central  position  a  dis- 
tance equal  to  the  lap,  and  if  the  automatic  cut-off  does  not 
interfere,  closes  again  at  the  same  point. 

Fig.  109  represents  the  Zeuner  diagram  for  the  Corliss 
engine  shown  in  Fig.  107,  and  the  marks  show  the  actual 


Steam 


FIG.  109. 

movements  of  the  steam  and  exhaust  valves  for  each  30 
degrees  of  their  movement  during  the  acting  portion  of 
movement. 

157.  Proportioning  Parts. — The  areas  of  the  steam-ports 
can  be  determined  as  already  shown  under  plain  slide-valves, 
and  the  exhaust-ports  should  be  from  i \  to  2  times  the  width 
of  the  steam-ports.  When  the  wrist-plate  is  in  its  middle 
position  the  steam-lap  varies  from  \  inch  in  the  smaller  sizes 


VALVE-GEARS. 

to  T\  or  ^  inch  in  the  larger,  and  the  exhaust-port  is  open 
from  T^  to  -J  inch,  depending  on  the  size  of  the  engine. 

The  point  o  is  generally,  although  not  necessarily,  in 
the  centre  between  the  four  valves.  The  point  d  is  on  the 
engine-frame  as  near  the  bottom  as  it  can  be  placed,  that  the 
points  c  and  e  may  move  as  nearly  as  may  be  parallel  to  the 
line  of  motion  of  the  engine.  The  length  of  de  should  be 
such  that  the  point  e  swings  equally  above  and  below  the 
line  O0,  and  when  this  is  the  case  the  distance  from  O  to  d 
horizontally  should  be  equal  to  g,  the  length  of  the  eccentric- 
rod.  An  eccentric-rod  extending  from  the  eccentric  to  the 
wrist-plate  would  be  inconveniently  long,  and  would  require 
bracing  to  keep  it  stiff  enough,  and  the  two  rods  be  and  ef, 
of  practically  equal  length,  are  substituted.  The  length  de 
is  sufficient  to  bring  the  hook  f  at  a  convenient  height  for 
handling,  and  otherwise  nothing  is  gained  by  making  de  or 
#/ greater  or  less,  as  it  would  be  possible  to  design  a  gear  in 
which  the  connection  from  b  to/" should  be  made  by  one  rod 
only  and  give  exactly  the  same  distribution  of  steam. 

The  throw  of  the  eccentric  varies  from  3^  to  10  inches, 
according  to  the  size  of  the  engine,  and  the  distance  of  is 
from  10  to  12  inches.  The  diameter  of  the  steam-valves  may 
be  made  about  \  the  diameter  of  the  cylinder. 

Referring  to  the  equation  giving  the  movement  of  the 
valve,  the  only  other  parts  to  be  determined  are  ql  and  of. 
If  in  this  equation  we  put  the  value  of  tf,  which  from  the 
valve-diagram  gives  the  proper  lead  and  port  opening,  and 

make  x  =  lap  -f-  lead,  and  GO  =  o,  we  have  a  ratio  for  —, 

which  can  be  used  in  determining  the  length  of  the  remain- 
ing parts  of  the  gear. 


INDEX. 


A 

ART.  PAGE 

Admission,  Movement  of  piston  during,       .        .        .        ,         ,       17  15 

"          Period  of •        .       17  14 

"          Point  of, .       12  10 

Advantages  and  disadvantages  of  radial  gears,      .        .        .         ,111  121 

Allen  link-motion,         .........       72  85 

"              "           Equation  to  movement  of  valve,      ...       73  85 

"              "           Valve  diagram  for, 73  85 

"      valve, .         •         .       27  2& 

Angstrom  gear, 109  120= 

"            "     Diagram  for, .  no  121 

Angular  advance,           .........       13  n 

"              "        Changing,  with  a  Stephenson  link,  ...       56  63 

"               "         Effect  of  changing 25  25 

"               "         with  gridiron  valves,         .         .         .         .         .122  132 

"               "         Regulating  by  changing,           ....       85  98- 

Armington  and  Sims  governor,     .......       89  101 

I"             "       "      valve, 28  30 

B 

Balanced  slide-valve, •        .       27  28 

Ball  engine  governor, •         .      90  104 

"         "      valve 91  105 

Bar  links = 44  5<> 

Bilgram  diagram,           .........  149  157 

"        Problems  solved  by  the,  .....  150  158- 

Bridges,  Thickness  of, -       35  4<> 

Buckeye  engine  eccentrics  and  connections,           ...  136  143 

"             "     Movement  of  valves  in,                  .        .        .  137  144 

'             "     Varying  cut-off  in, 139  145 

"     governor,      ........  14°  I4^ 

"    valve 135  143 

171 


3/2  INDEX. 

C 

ART.  PAGE 

•Centre  of  valve  travel,  To  lay  down  the,       •*.-».  59  67 

To  put  engine  on  the,         .......  38  45 

Changing  angular  advance  for  regulating,     .         «         .         ,         .  85  98 

"         eccentricity  for  regulating,    ......  86  99 

Circular  diagram  f  ^r  piston  position,    ......  34  37 

Comprersi  n  and  exhaust,  Equalizing,           •         •         •         •         •  33  36 

Movement  of  piston  during,       .         .         .        .         •  17  15 

P-riod  of, 17  I4 

Crossed  rods, •        •        •         •  55  61 

Curve  of  centres  for  Stephenson  link,  ......  47  56 

Cut-off,  Arrangement  for  varying,  with  gridiron  valve,        .         .  124  135 

"       blocks  of  Meyer  valve, 142  148 

"       "       "          "      Length  of, 144  151 

"       Equalizing, 31  34 

"       Equalizing,  with  radial  gears,  ......  104  115 

"    Stephenson's  link 57  65 

"       in  Buckeye  engine.  Varying,    ......  139  145 

'*      Limits  of,  with  gridiron  valves,        .         .        »        .         .  120  131 

"              "            "     Polonceau  gear, 133  140 

"       Point  of,                    12  n 

"       valve  diagram  for  Buckeye  gear,      .         .        .         .        .138  145 

"       Width  of  gridiron,          ..•••.  125  135 

"       with  gridiron  valve,  Varying,   ......  123  133 

D 

Designing  a  double-ported  valve, 26  28 

Fink  motion,     ........  82  94 

Gooch  motion,           .......  71  82 

"           Hackworth  gear,        .......97  no 

"          Marshall  gear, 108  119 

Meyer  valve,      ........  143  150 

"           plain  slide-valve, •        •  35  40 

"             "            "        approximate  solution,        .        .        .36  41 

"             "            "        with  equalizing  lever         .        .        •  37  43 

"          Stephenson  motion, 50  58 

Distribution  of  steam  from  the  diagram,       .        .        .        .        .  16  14 

Double  valve,        . .  115  127 

ported  valves, 26  28 


Eccentricity,  Average  valves  of,    .......       35  40 

Effect  of  changing,  .......       25  25 

"  Regulating  by  changing, 86  99 


INDEX.  173 

ART.  PAGE 

Eccentric  rod,  Length  of, 51  58 

"       Setting  the, 39  45 

•'      The, 3  2 

"         "     virtual, 49  58 

"       To  determine  the  position  of  the, 24  24 

Eccentrics  and  connections  of  Buckeye  engine,     .        .         ,         .  136  143 

Elliptical  diagram, 153  163 

Equalizing  bell-crank  leyer,           .......  32  35 

"          cut-off, •        .        .  31  34 

"                "      and  lead, 32  35 

"       with  radial  gears, 104  115 

"         "     Stephenson's  link, 57  65 

"          exhaust  and  compression,    ......  33  36 

"           lead  with  Stephenson's  link,         .....  56  63 

"           port  opening  with  radial  gears, 103  113 

Erie  engine  governor, 88  loo 

Error  of  the  Zeuner  diagram  for  the  Joy  gear,      .         .         .         .114  124 

"      Marshall  gear,      .        .        .106  117 

"        "           "             "            "      Stephenson  gear,  63  70 

Exhaust  and  compression,  Equalizing, 33  36 

Exhaust  lap, 5  4 

"        lead, 14  12 

"        Movement  of  piston  during,    .        .        .        .        .        •  17  15 

"        passages, I  i 

"        Period  of, 17  14 

"        ports,  Area  of, 35  40 

"      Width  of, 35  40 

Expansion,  Movement  of  piston  during, 17  15 

"          Period  of, 17  14 

F 

Fink  motion, .  74  87 

u       Designing  a, .         .  •  82  94 

"       Hanger  for  radius- rod  with  a, 80  92 

'           "       Lead  with  a, 75  87 

*           "       movement  of  valve  with  a,  Equation  for,           .         .  77  89 

*'          "       Radius  of  link  of  a, 75  87 

"          "       Setting  the  eccentric  of  a,          .....  81  92 

"      Suspension  of  link  in  a,     ......  76  89 

"          "       Valve  diagram  for  a,          ......  78  91 

Four  valves, 30  31 

G 

Gonzenbach  valve  (see  Gridiron), 116  127 

Gooch  link-motion 64  75 


INDEX. 

ART.  PAGE 

Gooch  Link-motion  applied  to  Polonceau  valve,  ....  131  139 

"  "  Designing  a, 71  82 

"  "  Lead  with  a,  . 66  78 

"  "  Movement  of  valve,  equation  for,  65  76 

"  "  Radius  of  link  in  a,  .....  67  78 

"  "  Suspension-rod  for  a, 68  79 

"              "            The  hanger  for  a,           .....  69  80 

"               "             Valve  diagrams  for  a,    .         .         ...         .70  80 

"  "  with  unsym metrical  parts,  ....  146  152 

Governing  by  changing  angular  advance, 85  98 

"  "  eccentricity, 86  99 

"  "  "  and  angular  advance,  .  .87  99 

Governor,  Armington  and  Sims, 89  101 

"  Ball  engine,  .  ....  ...  90  104 

"  Buckeye  engine,  c 140  146 

"  Erie  engine, .88  100 

Governors,  Throttling, 84  98 

Gridiron  valves,  .  116  127 

"  "  Angle  of  advance  with, 122  132 

"  "  Arrangement  for  varying  cut-off  with,  .  .  124  135 

'  "  Combined  diagrams  for,  .  .  .  .  119  130 

"  "  Diagrams  for, 118  129 

"             "       Limits  of  cut-off  with,           .....  120  131 

"             "       Varying  cut-off  with,    ......  123  133 

"            "            "       width  of  cut-off  valve,    ....  126  135 

"             "       width  of  cut-off  valve,           ...                  .125  135 

"  "  Width  of  ports  with,  ......  121  132 

Guinotte  gear, 146  152 

"  "  Movement  of  valve  with, 147  153 

"  "  Valve  diagram  for,  ...••,  148  154 

H 

Hackworth  gear, •  •  93  107 

"  "  attaching  valve-stem  outside,  .  .  ,  .  102  113 

"  "  Connecting  vp  a,  .  .  •  101  113 

••  "  Designing  a,  .....  ,  97  no 

*'  "  Equation  to  movement  of  valve  with,  .  .  95  108 

"  "  Error  in  Zeuner  diagram  for,  .  .  .  .  99  in 

"  "  Valve  diagram  for,  .  .  ...  96  109 

"  "  Variation  in  port  opening  with,  ...  100  113 

Hanger  for  radius-rod,  Fink  motion,  .  .  «  .  .  80  92 

"  "  Gooch  motion, 69  80 

••  Stephenson's  link,  .  •  •  53  59 


INDEX.  175 

ART.  PAGE 

Joy  gear,  The, 112  121 

"       "     Equation  to  movement  of  valve  with,    .        .        ,        .  113  123 

L 

Lap, 5  4 

"     Average  values  of, •        •  35  4° 

"     different  on  the  two  ends  of  valve,        •         .        •        .         .  31  34 

"     Effect  of  changing, ,..25  25 

Lead, 14  12 

"     and  cut-off,  Equalizing,         .......  32  35 

' '     Average  values  of,        ........  35  40 

"     Equalizing,  with  Stephenson's  link, 56  63 

"     with  Fink  motion, 75  87 

"        "     Gooch  motion,      .......         »  66  78 

"        "     radial  gears, 94  108 

Length  of  eccentric  rod,         ...••••.51  58 

"       "  link,      .........*  52  59 

"       "  valve  stem, 51  58 

Link  block,  Slip  of, 42  48 

"     Length  of, 52  59 

"     motion,  Allen, 72  85 

Fink, 74  87 

"        Gooch, •        »        •        •  64  75 

"           "        Reducing  slip  in  a,           ......  62  70 

*f           "        Stephenson's, 40  47 

"        To  determine  centre  of  suspension  of  hanger,          .  60  68 

"           "        To  lay  down  a, 58  67 

"           "        To  lay  down  centre  of  travel  of  valve,    .         •        .  59  67 

Link  motions, •        •         ,  40  47 

"     Radius  of  Fink, 75  87 

"              "       Gooch,           ........  67  78 

"       Stephenson's,        .......  43  50 

"     Stephenson's,  Point  of  suspension  of, 41  48 

Links,  Kinds  of •         .  44  50 

M 

Marshall  gear, 105  117 

"     Designing  a, 108  119 

"     Equation  to  movement  of  valve  with  a,          .         .  105  117 

"          "     Proportions  of,       .......  107  119 

Meyer  valve, 141  148 

"         "      Changing  distance  between  block  of  a,     ...  142  148 

"         "      Cutting  off  with  inside  edges,  .         .         .         .         .  I4S  15* 


1/6  INDEX. 

ART.  PAGE 

Meyer  valve  Designing  a, 143  150 

"         "      Length  of  cut-off  blocks  of,      .....  144  151 

Movement  of  upper  end  of  hanger,  Stephenson's  link,           .        •  53  59 

Movement  of  valve,  Equation  for  the,  with  Buckeye  gear,    .         .  137  144 

"              "           "     Fink  gear,          .         .  77  89 

"      "    "  Gooch  gear,   .    .  65  76 

"      "      "    "  Hackworth  gear,   .  95  108 

"  Joy  gear,  .    .    .113  123 

"                "               "              "           "     Marshall  gear,           .  105  117 

"                "               **             "           "    single  eccentric,  9  7 

««               ««               "             "          "    Stephenson  gear,  45  52- 

O 

Opening  of  port, 12  10 

Open  rods, 55  61 

Over-travel, 18  19 

P 

Piston  movement,  Circular  diagram  for,               .        .        •        „  34  37 

Piston  valves,        .......••.  28  30 

Point  of  suspension  of  link,           ...,.,.  41  48 

Polonceau  gear, 131  139 

"             "     Limits  of  cut-off  with, 133  140- 

"             "     Valve  diagram  for 132  139 

"          valve,           ......        o        ..  117  128- 

130  W 

"             "      Dimensions  of,  ..•••..  134  141 

Porter-Allen  engine,  regulating  a, 83  9$ 

motion,     .........  83  94. 

"                "       radius  of  link,       .«••••  83  95 

"           valves =  30  31 

Port  opening,  Equalizing,  with  radial  gears,         ....  103  113 

"          '*         variable  in  Hackworth  gear,  .....  100  113 

Ports,  area  of,       ..........  35  40 

"      Width  of,  with  gridiron  valve,   ......  121  132 

Position  of  stud  on  Stephenson's  link,           •        •        •        •        •  6l  68 

Put  engine  on  centre,  to,      ........  38  45 

R 

Radial  gears,         ...» Q2  107 

"         "      advantages  and  disadvantages  of,    .        .        .        .  in  121 

"      Angstrom's,     .....*...  109  120 

"    .  Definition  of  right-hand  rotation  with,     .         .         .98  ni 

"      Hackworth's, 93  107 


INDEX.  I?1? 

ART.  PAGE 

Radial  gears,  Lead  of  valve  with, 94  io& 

"          "      Joy's, 112  121 

"          "      Marshall's, 105  117 

Radius  of  link,  Changing,  for  Stephenson's  gear,           ...  56  65 

"           "        Fink  motion, 75  87 

"           "        Gooch  motion,      .......  67  78 

"           "        Porter-Allen  motion, 83  95 

"           "        Stephenson  motion, 43  50 

Reducing  slip, 62  70 

Relative  movement  of  two  valves, 127  137 

"       valve  circle, 128  138. 

"          "         "      To  draw  the, 129  138 

Rculeaux  diagram, „  151  160 

"              "        Solution  of  problems  by  the,    .         .         .         .152  161 

Reverse  lever,  effect  on  movement,       ......  23  24. 

"           "       To  design  equalizing,     ......  37  43. 

S 

Setting  the  eccentric, 39  45 

"         "         "          with  a  Fink  motion, 81  92 

Slip  of  the  link-block, 42  48 

"     Reducing 62  70 

Steam  inside  the  valvj,  Taking, 29  30 

"      lap, 5  4. 

"      passages r  i 

"      ports, I  i 

"     Area  of,    .                  35  40 

Stephenson  link-motion, 40  47 

"                  "             Changing  radius  of  link,          ...  56  65 

"                  "            Curve  of  centres,     .....  47  56 

"                  "             Designing  a, 50  58 

"             Equalizing  cut-off  with  a,         ...  57  65 

"                  "             Equalizing  lead  with  a,    .         .         .         .56  63 

**                  **             Equation  to  movement  of  valve  with,      .  45  52- 

"                  "             Error  of  Zeuner  diagram  for,  63  70 

"                   "             Hanger  for  link  of,  53  59, 

"                  "            To  lay  down  a, 58  67 

'*                  "            To  draw  valve  diagram  for,    .         .         .48  57 

"                  "            Valve  diagram  for,           ....  46  55 

Stephenson  link,  Point  of  suspension  of,        .....  41  48 

"      Position  of  stud  on  a, 61  68 

"            "      Radius  of, 43  50 

"            "      suspended  at  bottom,  Hanger  for,      ...  54  60 

"            "                   "         centre  of  chord,  Hanger  for,          .  54  60 

"            "      with  crossed  rods, 55  6l 

Straight  link,         .                           72  85 


178  INDEX. 


PAGE 

Throttling  governors, .        .        .  84  98 

Trick  valve, •••,.27  28 

Two  valves, 30  31 

V 

Valve,  Allen  or  Trick, «...  27  28 

Armington  and  Sims, 28  30 

"       Balanced  slide,           ...••••.  27  28 

"       Ball  engine 91  105 

"      circle,  Relative, 128  138 

"          "       To  draw  the  relative,      .•••••  129  138 

"      D  slide i  i 

'"            "      method  of  action  of,      ......  2  2 

"             "      movement  of,        ......         .         .  7  5 

"      Designing  a  plain  slide 35  40 

"                    "             "            approximate  solution,  ...  36  41 

"                    "            "           with  equalizing  lever,  ...  37  43 

*'      diagrams, 9  6 

"            "          Bilgram,    ........  149  157 

*'             "          Buckeye  engine, 138  145 

Elliptical, i£3  163 

'*             "          Error  in,  for  Hackworth  gear,  .         .         .         «99  m 

"            "              "         "       Stephenscn  gear,            .        .        .  63  70 

"             ««              "         "       Joy  gear, 114  124 

••             "              "         "       Marshall  gear,        .         .         .         .  106  117 

"            "          for  a  given  engine,  To  draw,      ....  15  13 

"            "           "    Allen  motion, 73  85 

"             «'           "    Angstrom  gear, HO  121 

'*             "           "    each  end  of  the  cylinder,               .        .         .  ,  17  14 

"           "    Fink  motion 78  91 

«*             «'           "    Gooch  motion, 70  80 

«'              •«           "    gridiron  valves, Il8  129 

«*            "          "                "            Combined,   .        .        .        .119  130 

te            •«           "    Guinotte  gear, 148  154 

'*             "           "    Hackworth  gear,           ....         *  96  109 

"            '"'          "    Polonceau  gear, 132  139 

"            "           "    Stephenson  gear,           .....  46  55 

"           "          "              "               To  draw,          ...  48  56 

"             "         Reuleaux, .  151  160 

"             ««        when  5  =  o, 10  7 

«•             "         Zeuner, 9  6 

«*            «•             " ir  10 

*'      Dimensions  of  Polonceau, 134  141 

*'       Double 115  127 


INDEX.  179 

ART.  PAGE 

Valve,  Double  ported, 26  28 

Equation  for  movement  of  a 9  7 

"  "  with  angular  advance,  .  .13  n 

"  face, 4  3 

"  "  not  parallel  with  piston  travel,  ....  23  23 

"  for  Ball  engine, 91  105 

"  for  Buckeye  engine,  .  .  .  „  .  .  .  135  143 

"  Gridiron  or  Gonzenbach, 116  127 

"  gear,  Guinotte,  ........  146  152 

"  "  Polonceau, 131  139 

"  Meyer, 141  148 

"  Piston, 28  30 

"  Plain  slide, i  i 

"  Polonceau, .117  128 

130  139 

"  Porter-Allen,  ....  c  ....  30  31 

"  Position  of,  for  any  position  ot  piston,  ....  6  4 

*'  seat, 4  3 

"  Setting  the,  .  «  •  39  45 

"  stem,  Length  of, 51  58 

"  To  adjust  the, 39  45 

"  Velocity  of  movement  of,  154  t64 

Virtual  eccentric,  The,           .                 .                  ....  49  58 


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22 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


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OVERDUE. 


14  1935 


LD 


LD  21-100m-8,'34 


YC   i28c 


359448 


UNIVERSITY  OF  CALIFORNIA  LIBRARY 


